Theory of formal tools for acquiring knowledge

Part I

Prolegomena to a genuine theory of the formal elements and structures necessary to acquire knowledge

Predicables and Predicaments, Terministic Logic

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This document is will prepare us for laying down a genuine logic, a logic which, although ultimately based on Being, that is on metaphysics, will nevertheless show that the Substance-Accident Metaphysics, as laid down by Aristotle and St Thomas Aquinas, and revised by us in First Part of Website ,  is the correct metaphysics, insofar as the ontological constitution of individual beings is being studied (that is, as being an approach different from that of studying ontological layers [See for the latter, Fourth Part of Website ] ).
Being a theory of formal tools for acquiring knowledge, this logic presupposes that the objects of knowledge, whether mathematical entities or extramental things, exist independently of being known, and are as they are, independently of being known, and can in principle be known precisely as they are in themselves. That is to say, our logic presupposes a realistic theory of knowledge. This theory is discussed in the document Realistic View of Knowledge, in first Series of documenst in First Part of Website .

The logic to be developed here is quite different from modern mathematical logic. While it is based on traditional Aristotelian-Thomistic logic, we will be at great pains to emphasize an important aspect, which is only more or less implicitly contained in the mentioned traditional logic, an aspect, that is, by which it is, or becomes, a genuine logic, namely the aspect of intentionality. While we will discuss this later in great detail, we can here say the following about this aspect :  Logic is about the formal elements which make knowledge possible. The subject of such a logic is therefore these formal elements. The objects of knowledge can be everything :  mathematical entities and relations, material entities and relations, etc. And insofar as they are objects of knowledge they are called 'real'. And as being real they are contrasted with the mentioned formal elements or tools by which they come to be known. These elements are formal signs :  they are entities such that their whole being solely consists of being about something else, that is, they are not merely likenesses of the things they are about (signify), meaning that they are not just instrumental signs, but are natural signs (themselves pointed to by words, which are conventional signs). Because the being of these natural signs only consists in their signifying other things than themselves, they are purely intentional signs or (equivalently) formal signs. Logic, in the sense of intentional logic, now studies such signs and their structures and the relations between them.
Modern mathematical logic on the other hand, is quite different. In fact it is not a genuine logic at all :  It deals with the nature of relational patterns or structures taken in their widest generality. These relational patterns are thus objects of knowledge, that is, objects of intention. They are not intentions themselves. Indeed even the so-called uninterpreted formal systems of mathematical logic are non-intentional structures, as well as their possible interpretations (that is, formal systems, now supplied with meaning). It is not so that an uninterpreted formal system intends or signifies one or another of its possible interpretations. That is, the uninterpreted formal system is not a logical intention, signifying some interpretation of it. Also one or another possible interpretation is not a logical intention. The intentionality just as such is rather to be found in the precisely intentional relation of identity that holds between the uninterpreted formal system and one or another of its possible interpretations. Indeed, an uninterpreted formal system is, by definition non-intentional, because of the sheer absence of meaning. Where then are the logical intentions in this case? Well, when the system is interpreted, meanings are assigned to the symbols, and this attribution of meaning  is  the logical intention (for example :  " R  is [that is, means]  'congruent to' "). Although this will be discussed further down, we will expound it succinctly here, not by invoking an example of a formal system, but an example of a proposition involving a formalized (that is, generalized) relation R :

((x R y) ^ (y R z)) ==> x R z

( In words :  if x is related to y by the relation R, and if y is related to z by the relation R, then x is related to z by the relation R ).

In fact it is asserted that the relation R is transitive.
"((x R y) ^ (y R z)) ==> x R z"  is a generalized relational pattern, namely :

We here have three elements standing in a certain relation to each other. This is an objective pattern, which is fully able to exist as the core of some real relation whether (only) in the mind or in extramental reality. So it is not a logical intention. It can, however, be intended by a logical intention, namely by the subject-predicate relation (of identity) :

The relation R is transitive

Now we're going to interpret our general expression ((x R y) ^ (y R z)) ==> x R z. This we do by means of four logical relations of identity, that is, four subject-predicate propositions :

x  is  a line segment.
y  is  a line segment.
z  is  a line segment.
R  is  the relation "is congruent to"

So the interpreted formula now becomes :

If  line segment  x  is congruent to line segment  y,  and line segment  y  is congruent to line segment  z,  then line segment  x  is congruent to line segment  z.

Like the general version, also this interpreted version is a relational pattern that can exist either in the mind or in extramental reality. So it is not a logical intention. But although it is not a logical intention, it can be intended by a logical intention, and this latter is embodied in the four subject-predicate relations given above.
In fact the transition from uninterpreted (or incompletely interpreted) to interpreted is clearly a transition from the more general to the less general. And this transition embodies a logical relation, such as between genus and species, or between universal and particular. But neither the uninterpreted nor the interpreted relational pattern is a logical relational pattern.

As to the relation R, when it is interpreted as a relation of congruence it can be expressed as :

x  is  congruent to  y

where x and y are line segments. This relation is not the same as a relation of sheer identity, because the orientation of the line segments does not need to be specified (and thus the line segments are not necessarily identical).
This pattern, involving line segment  x  and line segment  y,  is a real relation in the sense of an objective pattern, provided we read the expression as follows :

We have a line segment  x  and a line segment  y,  and they are connected to each other by the relation of congruence.

Here we have expressed  "x  is  congruent to  y"  as a two-term relation. So despite the presence of the word  "is",  it is as such not a subject-predicate proposition, because a subject-predicate proposition is a 'one-term relation' (if one may express oneself in this way, as do the mathematical logicians). So it is just a relation of congruence, and a relation of congruence is what it is, independently of its being known. And indeed it has objective properties such as being transitive. So (when expressed as a two-term relation) it is not a logical intention.
We can, however, express  "x  is  congruent to  y"  as a genuine subject-predicate relation in three ways :  Either we say :

which is a one-term relation, and as such a subject-predicate relation,

or we say :

which is also a one-term relation,

or, finally, we can say that we have a so-called unity of order, namely (here) a set of two line segments. And of this unity of order (figuring as subject) we predicate :  "is involving congruence of its two elements" :

which is also a one-term relation,

And either one of these (three) does intend :

x  is  congruent to  y

taken as a two-term relation.

In this way we hope that we, at least in a preliminary way, have made clear the difference between a real relation or relational pattern ("real" in the sense of objective) and a logical relation that can intend such a relation or relational pattern. The subject-predicate relation is a relation of identity because it first separates that for which the subject stands from that for which the predicate stands, and then re-unites them again (in predicaton). And it is this feature of separating and re-uniting that is the very essence of the act of intending.
Further down we will return to this important subject.

Predicaments, Predicables, Terministic Logic.

If we want to clearly understand what is in fact asserted by substance-accident metaphysics, and to what kind of logic it is connected, it is instructive to see things for a while along nominalistic lines, that is, along the moderate nominalism of William of Ockham and his terministic logic.
The terministic logic of William of Ockham is based on a metaphysics that is different from substance-accident metaphysics. Every thing, according to this metaphysics, which does really exist at all is an individual. What we encounter are individuals and nothing more. So it does not assume the existence of 'essences' as distinguished from the supposits (individuals) possessing such an essence. Further, the accidents (if we do not count Ockham's theology-based exception of Quality) are just different ways to signify the same given substance (which is an individual). That we do sometimes express a given substance by one of its accidents is either a result of the fact that we cannot describe a given substance completely by the same one term in every circumstance, or (is the result) of our wish to emphasize a certain aspect of that given substance. So the Aristotelian categories are not different types of Being, but different ways of signification (of the same given substance).
According to Porphyry (Greek philosopher, 3rd century AD) qua fundamentality the Predicables come first, and then the Categories ( = Predicaments). But according to Ockham he has reversed the priority originally set by Aristotle :  it is the Categories that come first and only then the Predicables. That is, the Predicables presuppose the Categories. Let us follow the nominalistic considerations on this score.
The basis for demonstration and science should distinguish itself from, and must be prior to, the syntactical operations of discursive thinking. So demonstration and knowledge of necessary and universal principles must be functions of meaning or signification. Therefore the analysis of modes of signification, as has been carried out in the Categories, is more fundamental than, and presupposed by, the analysis of modes of predication, as is done in the doctrine of the Predicables (expounded in Aristotle's Topica).
The system of Categories (Predicaments) is a classification of terms of first intention, based on difference in the way in which terms signify things that are not terms. All this considered in abstraction from questions of existence or fact, and from truth or falsity of any propositions involving these terms. The truth of a proposition depends on the signification (meaning) of its terms.
The terms dealt with in the discussion of the Predicables are considered as elements of propositions.
The Categories express the relation of term to thing (and thus boils down to the study of the ways of signification). If we make this a function of predication (that is, if we make these ways of signification dependent on predication), then the relation of term to thing, and thus its meaning, depends on judgement. And because the  truth  of a proposition depends on the signification of its terms, it now (as a result of Porphyry's reversal) depends ultimately on the form of judgement.
As a result of Porphyry's reversal the term  ' animal ',  for instance, means the Essence, albeit still in an incomplete way (in quid incomplete), that is :  the determinable Essence (it can be further determined by, for instance, rational ).  This on the basis of the way a genus is predicated of a subject, for instance :  homo est animal. This is praedicare in quid (incomplete), it is the expression of the Essence (of a substance). It should, however, be such that the meaning of the term  ' animal '  is independent of predication and judgement. The meaning of the term  ' animal '  must follow directly from the signification (the way of signification) of this term. As such, that is, according to the way of signification (as established in the Categories) the term  ' animal '  is a concrete and absolute term ( = not a connotative term NOTE 1 ),  not a term of fact), because it belongs in the first Category. The term  ' animal '  (and also its abstract counterpart  ' animality ' )  directly signifies an individual nature (thus not a [universal] nature residing in something) and connotes nothing.
On the other hand, all other categories (or at least that of Quality) contain connotative terms. Such terms signify primarily some individual thing (because it is only individual things that exist) and secondarily ( = obliquely) a content.
So according to the classification of terms of first intention into the Categories, the term  ' animal '  is absolute and does not refer to a (formal) content that is in something else, but refers to an individual nature, which here means a nature that is not only individual, but is also an individuum. This means that  ' animal '  and  ' animality '  refer to the same thing, but  ' animality '  includes a necessary aspect :  It can be considered to be identical to :    ' animal qua animal '  (that is, the abstract term can be reduced to its concrete counterpart). The same goes for  ' human '  and  ' humanity '.
Porphyry includes the  species  into the set of Predicables. But an infima ( = ultimate) species can only be predicated of an individual (of which, by the way, there is -- in the context of epistemology -- no science).
And so, when making the ways of predication prior to signification, causes the meaning of (say)  ' human '  to be the complete Essence, and as such it is predicated (essentially) of an individual.
When we say that the specific term  (  ' human ' )  signifies individuals essentially (per se), without connoting or implying any contingent fact or circumstance, it would be correct, because then the individual's essence is indeed signified, that is, the individual nature is indeed signified. But by making the  species  a  predicable  to express the relation between the species and individuals, the species is (now considered to be) attributable to individuals by an act of judgement and hence it is not itself  a  sign  of individuals, but stands for them (represents them) only through a human act of attribution or synthesis. So  ' human '  (or any other specific term, such as  ' dog ' )  is then attributed to a contingent fact (for example Socrates), that is (according to the way of predication), the (complete) Essence is attributed to a given contingent fact. And because this contingent fact can change, it cannot be identical to that Essence, which means that the Essence must reside in that contingent fact. All this happens because of the failure to distinguish between  meaning  (signification) (as established by the distinction of Categories) and  attribution  (ways of predications) (as established by the distinction of Predicables), that is, the failure to distinguish between  things  and  facts.
The  species  can, it is true, be predicated of the individual, but not  per se,  because Socrates (as Socrates) only  happens  to be human.
Because Porphyry considers all terms that can be predicated of a subject as regards their meaning dependent on ways of predication, all these terms are then attributable, that is, all these terms become connotative. So also a given species or a given genus is then predicated essentially of some indeterminate subject, that is of merely 'a thing' or 'something', meaning that they are predicated in the same way as accidents are predicated of a subject. But, whereas an accident is qua content apart from the subject, the essence (as is meant by a given genus or species according to the way of predication)  is  what the subject in itself is, there is no content of the subject left. What is left is its contingent aspect, and this is the same as 'a thing', 'a something'. This latter then is the ultimate subject of predication, making the subject of science identical to that of metaphysics (which is about the thing qua thing). It is that  in  which  forms,  connoted by concrete universal terms (such as  ' man ' )  and signified directly by their abstract counterparts, inhere, as parts in a whole. The term  ' man '  then stands for :  something in which humanity inheres.
But, because a sign is posterior, qua sign, to that which it signifies, and since terms of second intention (which, among others, the Predicables are) are  signs  of terms of first intention (or of second intention), it follows that the understanding of the Predicables (that is, the understanding of the ways of predication) is posterior to the understanding of the distinctions between terms of first intention made in the Categories.
Truth and scientific knowledge are to be exhibited as functions of the grasp of  meaning  -- of acts of understanding what it is to be such or such an individual thing. But such grasp of essential individual nature is only possible if some determinate concepts are absolute and not connotative (or, in other words, if not all terms are connotative). This is because a connotative term stands for a fact, that is, a contingent state of affairs.
Universal concepts of substance (such as  ' man ',  ' animal ',  and the like) are according to Ockham, first of all related to individual sensible things as  signs  to things-signified, or as acts of understanding to things-understood. And only now, being what they (the universal concepts of substance) are, can become predicates to subjects.
If we reverse this, then a distinction inevitably arises between what the universal term signifies and the things denoted by any subject term of which it is predicable, because predication is attribution (of something to something else).
The problem of universals, according to Ockham's analysis, is a consequence of Porphyry's confusion of signification with predication, or of meaning with attribution. For Ockham, who holds that individual things (which are the only entities that truly exist in extramental reality) are understood by generic and specific concepts precisely because such concepts are natural signs (acts or habits of understanding) of such individual things, there can be no problem of a relation between that what is signified or understood by a generic or specific concept, and the individuals in extramental reality for which it can stand.
But for Porphyry, who holds that genera and species are attributable to individuals by an act of judgement which is a synthesis of distinct elements, a distinction between the individual and the essential nature attributed to it, corresponding to the distinction between a subject and a predicate, becomes inevitable. From this distinction between individual things and their essential natures, arises the problem of universals and the corresponding problem of individuation.
Ockham argues against the notion that genera and species, or "universal things" corresponding to them, exist outside the mind as  parts  of individual substances, in such manner that substances are "individuated" by something distinct from the essential nature signified by the universal substantial term. If generic and specific terms or concepts signify substances, they signify individual substances, because a substance is precisely a thing that exists and is one  per se,  as distinguished from an accident (which as such is a mere contingent fact), which exists, but does so  per aliud  (i.e. in virtue of something else) and is particular,  all of this by reason of the existence of substances. A substance is an individual thing, and an accident a particular fact as expressed by attribution -- facts occur, and are particular, by reason of the individual substances which give rise to them. There is no cause of individual existence, no principle of individuation, apart from substances.
A substantial term signifies an individual nature, that is, it is an absolute term. It does not merely connote a form or exemplary cause distinct from these individual natures. Such a term is thus not connotative. If it were connotative, it would not signify changeable things  per se,  nor will it be an appropriate middle term for demonstrating attributes (which latter indicate ways in which things are essentially changeable) of subjects. To clarify this, let's look to the following syllogism :

Man is capable of laughing
Socrates is a man
Therefore, Socrates is capable of laughing.

Here the term  ' man '  cannot refer to some abstract essence (which resides in something), because an abstract essence, of course, is certainly not capable of laughing. So the term  ' man '  must signify some individual thing. And this means that the term  ' man '  is not connotative, but absolute. It doesn't connote some form distinct from something containing that form. And this means that there is no parallel with the connotative term  ' white ',  for which we can say that it primarily signifies an individual thing and connotes whiteness.
So  ' humanity '  is not merely connoted by the term  ' man '  but directly signified. This in turn means that  ' man '  and  ' humanity '  signify the same thing, and this thing is individual, because we have shown that the term  ' man '  signifies an individual. So both  ' man '  and  ' humanity '  signify an individual nature. The only difference between them is that  ' humanity '  involves some aspect of necessity, because it is equivalent to  ' man qua man '.
So all terms of at least the category of Substance are absolute terms.
A substantial term thus signifies the essence, which here means the individual essence, so there is no need anymore for a principle of individuation  ( because, as we have seen, in the thing there is nothing more than its individual essence :  this latter does not reside in anything else, so the individual essence is at the same time an  individuum,  it doesn't need to be individuated).  What is not signified by such a term are the replaceable determinations, such as the tanning of Socrates after him having enjoyed a sunny summer. So what is signified by a substantial term such as  ' man '  is the carrier-only, and that is the historical individual.
MOODY, The Logic of William of Ockham, 1935 (1965 edition), p.92, note 2, says rightly :

Ockham's point is that the term  ' this man '  has not a more determinate concept (act of understanding) corresponding to it, and giving it its meaning, than the term  ' man '.  The word   ' this '  is demonstrative, not significative. It is in this same sense that St Thomas says that the intellect does not understand particulars directly. Thus, while a univocal universal concept signifies individuals, so that the things understood by the concept  ' man '  are individual men, we do not understand (italics mine) individuals by any distinct or formal individuating principle, precisely because the circumstance (italics mine) of an individual happening to be present to us in such manner that we can point to him and call him  ' this man '  instead of  ' a man ',  is not a part of the reason (italics mine) of the being of the individual. Though everything is per se one and individual, nothing is per se pointed to.

The predication of the species (as term of first intention) of a subject.
Ockham (See MOODY, p.95), following Porphyry, includes the  species  as a fifth Predicable (Aristotle only mentions four of them). Ockham differs from Porphyry, however, by holding that the specific and generic terms signify precisely the individual things for which they can stand NOTE 2 ),  per se  and non-connotatively. Consequently, his inclusion of the species as a fifth Predicable does not alter the Aristotelian thesis that the individuals of which a scientific proposition is true, and for which its terms stand, are signified  per se  and absolutely (that is, non-connotatively) by the generic or specific term, independently of particular fact and of the truth of contingent propositions. The species is a Predicable, because it can be a predicate in many singular propositions -- but it cannot be predicated  per se  or necessarily of any subject NOTE 3 ),  nor is it properly speaking a predicate in direct or 'natural' predication (and this could thematically persuade us not to recognize the species as a genuine Predicable).
For Ockham, as for Aristotle, the primary subject of all necessary and scientifically known propositions is the universal term, which is a genus or species.

Predicables and signification.
The division of terms according to the modes of predicability  p r e s u p p o s e s  the distinctions between modes of signification established in the Categories, and not, as Porphyry held, vice versa. Genera, differentiae, and propria are all predicated necessarily rather than contingently of their subjects. Their distinction thus rests not on the way in which they are predicable of their subjects, but on the way in which they signify the things signified  by  their subjects  ( In 'S is B', where B is a genus, or a difference or a proprium, B signifies, in respectively different ways, the things signified by S ).
A genus signifies the things for which it can stand,  per se  and non-connotatively. In  ' homo est animal ',  for example,  ' animal '  can be predicated as a genus (and not as a difference or as a proprium), not because of the way it is predicated of  ' homo ',  but because it signifies the things for which it can stand, namely the things signified by  ' homo ',  necessarily and directly, without connoting anything else. And, again, because of this it can be predicated as a genus.
An essential difference (differentia) signifies the things for which it can stand,  per se,  but connotatively, by indicating what they are only partly (it connotes the rest). In  ' homo est rationale ',  for example,  ' rationale '  can be predicated as a difference (and not as a genus or as a proprium) because it signifies the things for which it can stand, namely the things signified by  ' homo ',  necessarily, but connotatively, by indicating partly what the significatum of  ' homo '  is, namely rational, and connoting something which has rationality.
A proprium (property, necessary attribute) signifies the things it can stand for,  per aliud  (i.e. by something else) and connotatively, through connoting something extrinsic to them (i.e. to those things) as (this extrinsic entity is) potentially existent by reason of their individual nature (that is, the individual nature of these things) (MOODY, p.96). In  ' homo est risibilis '  (in the sense of man is able to laugh), for example,  ' risibilis '  can be predicated as a proprium (and not as a genus or as a difference), because it signifies the things for which it can stand, namely the things signified by  ' homo ',  by invoking something else, namely the ability to laugh, i.e. it primarily signifies these things, but secondarily the ability to laugh, which is (considered to be) extrinsic to these things. Here the signification is  per aliud,  but the predication is  per se.
An accidens, finally, signifies the things it can stand for, also  per aliud  and connotatively. Their distinction from propria becomes only evident in a predication, namely when this predication is  a  per accidens  predication, in virtue of the observed fact that they turn out to be without qualification replaceable determinations. An accidens as term of second intention (thus as a predicable)  is  a  term  which is contingently predicated of a subject, in such a manner that it may be affirmed or denied of the subject, with the latter standing for the same individual thing or things. On the other hand, an accident as term of first intention signifies an auxiliary being which contingently further determines a substance (that is this further determination of the substance does not pertain [and we must not say :  is not predicated] essentially to it).
Terms are predicable of many as genus or as species when they signify precisely the things for which they can stand in any proposition, without signifying or connoting anything extrinsic to the essential or individual being of what they can stand for, and without signifying such individuals through indicating a part of their nature in abstraction from their undivided being. Such terms are said to be predicable (in virtue of their signification) of a subject term  in quid,  being appropriate answers to the question  " what is this? "  pointing to something.
In a loose sense (MOODY, p.97), terms in the accidental categories are also used in answers to such questions, as we point to a black object and say :  "What is the black?" and receive the answer  " It is the colored ".  Only in this improper sense NOTE 4 )  do we call  ' colored '  the genus, of which  ' black ',  ' white ,'  ' blue ',  etc. are species. All these terms are concrete and connotative. The absolute ( = non-connotative) terms in the Category of Quality, however, like  ' whiteness ',  ' blueness ',  ' heat '  (molecular impetus), etc., are in the strict sense species of qualitative genera, such as color, or impetus (where Quality is the only accidental Predicament according to Ockham) NOTE 5 ).

We must realize that a given accidental predicament can be either a proprium or an accidens (that is, either belongs in the Predicable  ' Proprium ',  or in the Predicable  ' Accidens ' ),  as is, for instance, the case with the term  ' blue '.  Its subsumption (subalternation) under one or another Predicable depends on the predicative context. Because the term is connotative, its signification is either essential and connotative, or non-essential and connotative, and thus it can be predicated either as a proprium or as an accidens :
If we have the predication :

A copper sulfate crystal (with the chemical formula)  CuSO4 . 5H2is  blue,

then we have to do with a genuine  proprium  of  a  CuSO4 . 5H2O  crystal (albeit not a specific proprium, but a generic proprium, because also many other things are [intrinsically] blue, for example Cu(NO3)2 . 5H2O ).
The blueness of the   CuSO4 . 5H2O  crystal is a non-replaceable determination, or (equivalently) a phenotypic essential part of such a crystal.
If, on the other hand, we have the proposition :

My bruise [and thus part of me]  is  blue,

then we have to do with  an  accidens,  because a bruise can also be red, or can be totally absent.
The blueness of my bruise (and thus part of me) is a replaceable determination.

When a genus is predicated of a species, such as in  ' homo est animal ',  it is not asserted that the subject term is the predicate term, nor that the species is the genus or part of the genus (the genus is part of the species, but not literally so) -- but what is asserted is that each of the individuals of which the specific term is a sign, is signified also by the generic term. The terms of such propositions do not signify classes, nor anything distinct from the individuals for which they can stand, nor do they signify or connote something that is a part of these individuals. But both a genus and a species signify the whole individual  ( ' undivided ' )  nature of individual things, in abstraction from contingent circumstances under which they might happen to have been apprehended in sense experience. These contingent circumstances are the (replaceable) states or determinations of the individual thing (which here, as stripped of these replaceable determinations) is then a historical individual [to make sense of 'replaceable'] ).
See MOODY, p.98/99.

Porphyry's language, in his exposition of the predicamental order whereby species are subordinate to genera, and 'superior' to individuals, etc., is such that this predicamental order is frequently taken as an order of  things,  or of metaphysical 'parts' of things, outside the mind, instead of as an order of  terms  or intentions that are merely signs of individual things. For example, the statement so often made that the species is 'part' of the genus (as is suggested by comparing their respective extensions), or that the genus is 'part' of the species (as is suggested by comparing their respective comprehensions or contents), cannot be understood in any literal sense, either as applying to terms, or to things that are not terms.
The term  ' man '  (indicating a species) is not a part of the term  ' animal '  (indicating a genus), nor vice versa. Similarly the individuals signified by  ' man ',  -- Socrates, Plato, etc. -- are not parts of the things signified by  ' animal ',  nor the other way around. For  ' animal '  signifies Socrates, Plato, and any other individual man, and so does the term  ' man ',  but we cannot say that an individual thing is part of itself. See MOODY, p.99/100.

One sometimes says that the genus is a part of the definition of the species (that is, of a given species) (which is correct), and because the definition is the same as what it defines, the genus is part of the species. But this is false, because the definition is not the same as what it defines, except metaphorically. Although both the definition and the term which is defined by it are signs, they are different signs :  definiendum and definiens, for example :  homo and animal rationale. They are the same only in the sense that the definition is a sign of the same things that the term defined is a sign of  (MOODY, p.100).

Ockham's entire discussion of the ways of comparing genus and species (such as in  ' the genus is part of the species ' )  is directed against the tendency, common enough among medieval interpreters of Porphyry, to regard the predicamental order known as the  ' Tree of Porphyry '  NOTE 6 )  as a metaphysical order. It is, according to Ockham, a purely logical order, whose elements are not things in extramental reality, nor parts of such things, but intentions of the mind that are signs of individuals (that is, mental signs, pointing to extramental individuals).
It is not because the generic term  ' animal '  signifies something  in  Socrates and  in  Plato distinct from what the specific term  ' man '  signifies, that the one term is related to the other as genus to species. On the contrary, it is because  ' animal '  signifies, as an individual or undivided nature, each thing that  ' man '  signifies as an individual nature, ... that  ' animal '  is predicable universally of  ' man ',  and is a genus of  ' man '  NOTE 7 ).  And because  ' man '  cannot universally be predicated of  ' animal '  (we cannot say  ' animal is man ' ),  ' animal '  is the  genus  of  ' man '  (and not  ' man '  is the genus of  ' animal ' ).
The determination of how one term can stand for the same things that another term can stand for,  presupposes  the possibility of determining -- independently of knowledge of the truth or falsity of such predicative relations -- what the terms  can  stand for, and that is to  define  their meaning through the grasp of essential or undivided nature.

According to Ockham both a given genus and a given species signify the whole individual undivided nature of individual things in abstraction from contingent circumstances under which they might happen to have been apprehended in sense experience (MOODY, p.98/9). This means that of course also the highest genera, each for themselves in the sense of any given highest genus, signify the whole individual undivided nature of individual things in abstraction from contingent circumstances. Only this time the signification is maximally indeterminate. A Predicament is the most general answer to the question  "what is it?" (where  "it"  can be a real thing or a state of such a thing, or, equivalently,  "it"  refers to something in extramental reality). So we cannot agree with Ockham (and maybe even not with Aristotle for that matter) that the Categories (Predicaments) are just forms of discourse rather than kinds of things (basic types of things), or, perhaps better, basic kinds of entities, in extramental reality. They are, it is true, modes of signification, but these modes reflect what the things are, either in themselves or what they are accidentally. As for what they are in themselves :  Because, say, a lion is ontologically different from yellowness, lion and yellowness are signified in different ways :  The term  ' lion '  is concrete (while the term  ' lioness '  or  ' lionity '  is abstract, but here standing exactly for the same thing) and absolute ( = non-connotative), while the term  ' yellowness '  is abstract. And if we consider the concrete counterpart of  ' yellowness ',  namely  ' yellow ',  then the term is connotative :  It signifies primarily 'something' and connotes  ' yellowness ',  that is, it legitimates the attribution of yellowness to that something, resulting in yellow. So this difference in signification is dictated by a difference in ontology between a lion and yellowness, as they are in themselves.
So what things  in themselves  are, is reflected in the Categories by the distinction between Substance and Accident, that is, either signifying a given thing (in the sense of an entity) as a substance, which in turn means signifying it by a concrete and absolute term [such as  ' lion ' ], or signifying another given thing (in the sense of an entity) as an accident, which in turn means signifying it by either an abstract term [such as  ' yellowness ' ],  or by a concrete but connotative term [such as  ' yellow ' ].
But that does not exhaust the function of the Categories, because there we see not only a differentiation between substance and accident, but also a differentiation between the several accidental categories, that is, there are several distinct accidental categories. And this means that we can view the categories in a different way (more closely to the view of Ockham) :  We can hold that, in every particular case, all the ten Aristotelian Categories (Substance, Quantity, Quality, Relation, etc.) are terms signifying one and the same thing, but signifying this same thing in different possible ways, in different degrees of completion. So we can describe Socrates (taken as an example of human beings, not taken as Socrates) as to what he essentially is :  we then describe him as a substance. But we can also describe him as to his length :  we then describe him as a quantity. Further, we can describe him as to his skin color :  we then describe him as a quality. Again, we can describe him as to his being a teacher of Plato :  we then describe him as being related to, etc. And here we clearly see what Ockham means by holding that the Categories are different modes of signification (it is thus about different significations of a same given thing, which here is always an individual thing, in the sense of a genuine individuum, existing ontologically independently of other such things).

The species, that is, a given species, signifies in the same way as the genus :  It signifies, but now more completely, the whole individual nature of an individual thing, abstracting from contingent states. So we can say that a given species, as well as a given genus (which are natural signs) wholly from and by itself, that is,  per se,  signifies that what it means. However, in contrast to a genus, a species can only be predicated of a particular individual (a genus can also be so predicated, but it can also be predicated of a species [for example in :  ' homo est animal ' ] ).  Indeed, when we say  ' Socrates is a human being ',    ' a human being ',  which is a species, is predicated of a particular, not, however, predicated  per se  of Socrates, but  per accidens,  because Socrates is a contingent fact. To counter this we could perhaps hold that the term  ' Socrates '  here in fact means :  ' Socrates and Plato and Peter and ... '  But this is a collection of (all) human individuals which collection is not itself a human being. But perhaps we could hold that  ' Socrates '  refers to  ' Socrates or Plato or Peter or ... ',  but then we say nothing more than  ' a human being is a human being '.  So we are left with the finding that although the term  ' (a) human being '  signifies  per se  individuals in abstraction from contingent facts, it is always predicated  per accidens,  because it can only be predicated of particular individuals. And this is why the species is not a predicable within natural science, because the subject of the species (when the species is predicated) is a particular fact (indicated by a particular term, such as  ' Socrates ' ),  while this subject, as the ultimate subjects of science, should be something that is signified by universal terms (such as a species, which is the subject of which a genus can be predicated, or of which a proprium can be predicated).

The interpretation of the Categories.
The system of Categories lists the most basic kinds or types of things ot beings, I maintain.
But of course, as kinds (of things) they do not exist in extramental reality, because there only individuals exist. As such Ockham is right in holding that the Categories do not list (extramental) things. The real reason, however, that Categories are, according to Ockham, not (extramental) things, is, as already stated above, that Categories are ways of signification always of one and the same thing (which is always a substantial thing), implying that they do not signify more than one different thing, that is, where "more than one different thing" means that each category would signify a different kind of thing : 
Socrates is a human being (characterizing him as [i.e. by means of] being an instance of Substance).
Socrates is 1.70 m long (characterizing this same Socrates as being an instance of Quantity) NOTE 8 ).
Socrates is a father (characterizing the same Socrates as being an instance of Relation).
Socrates is white (characterizing him as being an instance of Quality).
So these are all kinds of ways to describe (for example) Socrates (in degrees of completeness and essentialness), that is, only Socrates, and not (to describe) other things (such as quantities, qualities, relations, etc.). Therefore, the Categories (Predicaments) are, according to Ockham, not different (extramental) things, but ways (modes) how to describe one and the same (substantial) thing (See MOODY, p.131--132).

Of course we can ask :  How it comes that (any) one and the same thing can be described in a number of fundamentally different ways?
According to Ockham the terms in the Categories are signs of things and consequently distinct from the things of which they are signs. But this is of course the case with respect to every natural sign (that is, also with signs signifying signs which are not extramental things).
A Category, such as Substance, Quantity, or Quality, signifies as a genus (in this case as a supreme genus) a thing in the most general, and therefore indeterminate way (but not totally indeterminate such as the term  ' something '  does). This we find when determining the  ' what is it? '  of a given thing in the most general way. And because many (but not all) other things so (generally) determined as to what they are, end up at the same Category or supreme genus (for instance Quality), this Category is then one of the basic types of things or beings. As  a  type  it does not exist, as has been said, but it signifies many individual things disjunctively.
So (on the basis of our own findings) we see that Ockham not totally succeeds in demonstrating that a category is not signifying an ontological aspect of a thing (and another category signifying another ontological aspect of the [same] thing). It is difficult to assert that, for instance, Quality is convertible with Substance and convertible with Quantity and with all other Categories (while UNITY [as one of the agreed-upon transcendentals] can be said of any substance and of any accident, involving therefore all categories, Quality does not intrinsically involve any other category, and also each of the other categories does not intrinsically involve any other category, implying that there is a true difference between genuine transcendental terms [such as UNITY ]  and accidental terms [such as Quality, Quantity, etc.] ).
The fact that a same substantial thing allows itself be signified in a number of fundamentally different ways, is, I venture to say, indicative of such a thing actually possessing a corresponding number of different ontological aspects, whitout thereby considering these aspects as different things in their own right.

The term  ' predicament '  (or  ' category ' )  is used in two senses :

The genus generalissimum  as  genus generalissimum  is a term of second intention which stands for terms of first intention, in the sense that the genus generalissimum says something of a genus (term of fist intention, for example  ' Quality ' )  namely that it is a highest genus (i.e. one of the highest genera). So a given genus generalissimum (and now not as genus generalissimum) can figure as a term of  first  intention, as we can see in  ' Quality '  which signifies  ' something in which quality is present '  (nominal definition of the term  ' quality '  --  there is no real definition of  ' quality ',  likewise as it is in the case of all the other Categories). So the genus generalissimum can refer to a thing in extramental reality. The  residing-in  of quality is, according to Ockham, connoted by the term  ' Quality '.  And this same thing (the 'something' mentioned earlier) can be signified by another Category, for instance  ' Quantity '  or  ' Substance '.  Only in the case of Substance, the term  ' Substance '  does not connote anything. All other categorial terms, according to Ockham, do connote something non-substantial.
But maybe what is connoted and what is primarily signified by an accidental category is merely stipulated (by Ockham) in order to fit things well into the thesis that the Categories do not refer to ontological aspects :
We could, for example, say that in the case of  white  (and so also with respect to all other terms belonging to one or another accidental category) the term  ' white '  primarily  signifies whiteness and merely connotes that this whiteness must be present in some appropriate substrate or carrier (logically :  subject). If this is correct, then Substance and Quality signify  different entities  in extramental reality, that is, entities that are formally totally distinct, as a carrier-only and a replaceable determination. And, I might add, also with all the other accidental categories. And then the categories are not mere ways to describe a particular thing, but signify different entities existing in extramental reality.

Aristotle (MOODY, p.141) says that Substance has no contrary. This characteristic, as applied to  terms  in the Category of Substance, is revealed by the definition of contrary terms -- i.e. terms are contraries when they cannot be predicated of the same individual subject, or singular term (such as  ' Socrates ' ),  simultaneously, though they can successively.
But every subject of which contrary terms are predicable stands for something for which the term  ' substance '  can stand. And since contrary terms cannot be predicated of each other (we cannot, for instance, say :  blue is red), it follows that every substantial term, though it can be a subject of contrary predicates, is itself the contrary of no predicate (when we, for instance, say :  a dog is a mammal, or, the dog Fido is black,  ' dog '  and  ' mammal '  (and also  ' Fido '  and  ' black ' )  are not contraries, because the one can be predicated of the other).
It is true, likewise, that substances in extramental reality have no contraries (which is probably what Aristotle had chiefly in mind), for the individual is  per se  one, and every substantial term signifies individuals  per se  (that is, signifies the individuals directly and connotes nothing), so that coordinate species of substance (for instance MAN and DOG) cannot signify a single identical thing (with the one or the other contrary as residing in a subject) either at the same time or successively (MOODY, p.141), which latter implies that they cannot replace each other in a process (recent species [at any one time] did not, and cannot descend one from the other, but they can have a common ancestral species). So substances are not distinguished from each other as contraries.
But, I might add, then substantial change would be impossible, because the possibility of substantial change implies the existence of an ultimate, but totally undetermined (as regards formal content), substrate, which then is a subject for contrary (substantial) predicates (correponding to substantial forms).
Indeed, it is, because of Aristotle's statement that substances are not differentiated from each other as contraries, that Ockham denies the existence of prime matter as ontological constituent of a (given) substance. Indeed, the term  ' substance '  only logically has a subject, as in  ' Socrates is a substance ',  but ontologically it is not carried by anything, but carries replaceable determinations.
However, the possibility of substantial change (versus mere accidental change) must be conceded :  An Andalusite crystal (to take just an example) can change into a Kyanite crystal when thermodynamic circumstances change in a certain way :  The Space Group symmetry  Pnnm  is then replaced by the Space Group symmetry  P1*  [where * is usually notated as a score above the numeral] (while the chemical composition remains the same :  Al2SiO5 ),  and this is a true substantial change :  The Essence  Pnnm + Al2SiO5  is replaced by  P1* + Al2SiO5  ( In the document Crystals and Metaphysics in the first Series of documents in First Part of Website we have found that the Essence of a given crystal is its Space Group symmetry PLUS its Chemical composition, or at least it is the phenotypical expression of this Essence). And a replacement of one Essence by another is equivalent to the replacement of one substantial form by another such form. And these forms seem to be true contraries. They exclude one another, but can occur successively (but see below).
From this it is clear (and even more so, when also the chemical composition is (in relevant cases) replaced by another, or when we have to do with nuclear changes [one chemical element turns into another element] )  that there must be an ultimate substrate (otherwise such changes were not changes, but annihilations in nihilo and subsequent creations ex nihilo), and this substrate is prime matter. Now we can indeed predicate  ' Andalusite '  of this substrate :  ' This (entity) is Andalusite ',  and later (after the thermodynamic circumstances have changed) we can predicate  ' Kyanite '  of this substrate :  ' This (entity) is Kyanite '.
But are  ' Andalusite '  and  ' Kyanite '  contrary terms (signifying contraries)? Superficially, yes, but in fact they're not.  ' Andalusite '  and  ' Kyanite '  cannot, it is true, be predicated simultaneously of a subject, that is, of the same subject. Seen this way, they should be contraries. But in the case of true contraries (as defined above)  ' the same individual subject, or singular term '  signifies a subject that is the same, not only qua individual, but also qua formal content. That is, any given formal content of a subject (substrate) is such that it cannot sustain contrary determinations at the same time (say red and blue). But prime matter can do so :  Because prime matter is not of itself partitioned into contiguous parts, it is  per se  one. And this one substrate can sustain 'contrary' forms simultaneously as long as they are spatially discrete. It is only in virtue of the  form  and all that it entails, that we can speak of  ' this substrate here '.  The ultimate substrate as such can be informed simultaneously by 'contrary' forms, that is by forms, which would otherwise be contraries.
So these forms (substantial forms) are not contraries.
It is important to realize that the ultimate substrate, which is the materia prima, is not a thing. It is not a 'receptacle' (of forms), nor a void, nor anything like that :  Prime matter cannot exist on its own :  it is the form, albeit only in a potential way.
Thus we have established that in the substantial order we do not have to do with contraries (because they can exist simultaneously in the same subject (ultimate substrate, prime matter)). Only in the order of accidental change can we have to do with contraries (that is, there we have to do with a certain type of different formal contents that cannot simultaneously inhere in the same subject (substrate)).

The status of the accidental categories.  Preliminary framework.

We ask ourselves :  is  ' quantity '  an accidental term (i.e. a term adding something to substance when predicated of it, like the term  ' white ' ), or, as Ockham wants it to be, a transcendental term (that is a term that is adding nothing to substance when predicated of it, like the term  ' ONE ' ) ?
As regards  ' being ,'  it is agreed by everyone that this is a transcendental term (when we say  ' Socrates is a being ',    ' being '  does not add anything to Socrates.
But, although  a  lion  is a being, the term  ' lion '  is not a transcendental, because not every being is a lion (while every being is ONE).
The same with quantity. If the term  ' quantity '  is transcendental, then its species, say  ' 1.70 m length '  need not be transcendental, because, indeed, not every being is 1.70 m long (whereas every being is ONE).
If  ' quantity '  is a transcendental term, this means that it transcends all classifications of Being, especially the Categories. However, we cannot say that every being is (a) quantity -- as though  ' quantity '  would transcend every classification of being -- (like we can say that every being is a unity, and thus  ' unity '  transcending every classification of Being). We cannot say this without qualifying it. So indeed, the assertion

Every being is (a) quantity

can, as such, not be true if it were to mean that every being is nothing more than (a) quantity. So we must rephrase it as follows :

Every being is quantitative

(in the sense of quantitatively determined). Now it leaves open the possibility that, in addition to being quantitatively determined, it can also be determined in other ways.
So far so good.
But if every accidental category turned out to be transcendental, then we have lost the classification of Being that is to be transcended by transcendentals. And so there would be no transcendentals after all. But the Categories were supposed to be the basic types of things or beings (to which correspond basic types of terms of first intention, that is, basic types of signification). If this classification is gone, then the next classification layer becomes the most basic classification of Being. And indeed, when there are only individual substances in extramental reality, then the most basic classification is just the classification of  species  of Substance. This classification we see in chemistry with respect to atoms, molecules (which both are individual substances) and single crystals (which also are individual substances), and in biology we see it in the taxonomic systems of species of living beings.
So if we speak about :  ' applies to all beings ',  then we must, in the context of an amended Ockhamistic philosophy, think of these classifications (which are classifications of species of Substance), together forming the one classification of all beings.
And when we now ask whether a term is transcendental, we must ask whether it is predicable of (at least) every being that is mentioned in our basic classification.
Now, the term  ' quantity '  refers, not to one or another being, but to some aspect of something. We cannot directly predicate  ' quantity '  of a (given) substance, a thing, a being  (although we can predicate it directly of something as long as we remain within the proper domain of quantity :  we can say :  length is a quantity ).  We can only predicate it directly of a substantial being when we transform the term  ' quantity '  into its concrete counterpart :  ' quantum ',  and then we can indeed say  ' Socrates is a quantum '.  However, this is not fully satisfying because it seems still to mean that Socrates is nothing more than a quantum. So indeed the best way to predicate this category of a being is :

Socrates is quantitatively determined

(suggesting now a little better that this is not the only way he is determined). And from here we can say that  ' quantitative '  transcends the most basic classification of Being (as established above), because every being whatsoever is quantitative (if we only and exclusively consider the domain of material extramental reality). And we will call  ' quantitative '  not already  "transcendental"  when it transcends the basic classification, but only if, in addition to this, the term  ' quantitative '  necessarily signifies primarily the substance, the substantial being, and only connotes the quantitative aspect, that is, not  adds  it NOTE 9 ).  With this we mean, accordingly, that something being quantitative does not mean that anything has been  added  to the being, resulting in it (now) being quantitative, as it was asserted by certain philosophers.
It is this  addition  that is denied by Ockham.
But this must be demonstrated.
We can still say :  To every being, something is added, which itself is not a being, namely quantity. Indeed, if we acknowledge -- with Ockham -- that only individual substances exist in extramental reality, then quantity is not a being whatsoever. But when something is not a being, it is nothing (in extramental reality). So nothing is added to being when it is conceded that it is quantitative. Accordingly, this means that  quantitative  is nothing more than a term. A term, signifying not a being apart from the only beings there are, namely individual substances, but signifying these individual substances themselves. And it is in this way that we call  ' quantity '  (  ' quantitative ' )  a transcendental predicate, just like  ' being ',  and  ' unity '  are transcendental predicates. And the same can be argued for the other  ' accidental '  predicates, such as  ' quality '  (  ' quality '  was not considered to be a transcendental by Ockham, but this principally because of theological doctrinal reasons, which we can dismiss here, and so letting  ' quality '  to be a transcendental as well).

So we see that Okham's assertion that the accidental categories are in fact transcendentals, is based on the basic premise of his philosophy, namely that in extramental reality only individual substances exist, that is, the only beings are individual substances. From this premise it follows that quantity, quality, etc. do not exist in extramental reality -- they are not beings -- and so they cannot be something that is added to beings. Their names are terms signifying directly these existing individual substances.

The term  ' quantity '  is predicable universally and  per se primo modo  (as a definition is  per se  predicated of its definiendum) (See Part IV, NOTE 145a )  only of those terms in nominal definitions of quantitative states in which it (i.e. quantity) is included. For example :
The nominal definition of the term  ' 1.70 m long '  (which signifies a quantitative state) is :  Something in which  1.70 m length  is present.
And indeed, the term  ' quantity '  can be predicated  per se primo modo  of the term  ' 170 m length '  :  1.70 m length  is  a quantity.  This predication is possible because the definition of  ' 170 m length '  is :    ' a  length  that ... (specifying this particular length using a standard to which one can point) ... ',  and this definition ultimately leads to the term  ' quantity '  as soon as we're going to complete it by  ' climbing up '  (the ladder of generality, starting)  from  ' length '.  Indeed, by doing so, we end up at the term  ' quantity ',  now including all genera and differences (differentiae) lying between  ' length '  and  ' quantity '.
But of course any  infima species  (that is, ultimate species, species specialissima) of quantity, say,  2 cc volume  (or  1.70 m length  for that matter) cannot be universally predicated of any abstract quantitative term :  We cannot say :

5 cc volume  is  2 cc volume,


1.70 m length  is  2 cc volume,

because  2 cc volume  differs from  5 cc volume,  and from  1.70 m length.
Nor can we say

quantity  is  2 cc volume,

because not every quantity is 2 cc volume.
An infima species of quantity can only be predicated of a substance. And then only when this species is expressed by a concrete term (for instance :  Socrates is 1.70 long), but then only contingently so, if, with respect to the latter, we follow Ockham  ( In his Expositio Aurea, II, 58 v, Ockham says [MOODY, p.151, note 1] :  "quamvis quantitas non sit alia res a substantia et qualitate, tamen contingenter predicatur de substantia." [translation :  although quantity is not considered to be something different from substance and quality, it is nevertheless predicated contingently of [a given] substance] ).
But is a quantitative term in all cases predicated  c o n t i n g e n t l y ?  At least there are many such occasions. For example in the situation expressed by  ' Socrates is 1.70 m long '  (We can say :  At the time of observation Socrates happened to be 1.70 m long). But are there also other cases, fundamentally different in this respect?
In regard of this we must consider how things are when we have to do with a quantitative term signifying for example the size of a given atom belonging to a certain atomic species, which size is, however,  intrinsic  to that (here, the mentioned atom) which has this quantity. Indeed, the size of, say, an oxygen atom must be predicated  per se  of that atom (as representing a species), and thus not contingently so. Let me explain. An atom of a given atomic species necessarily has a definite size, a size, equal to that of all members of that species (under the same circumstances). Therefore, the size of a given atom is a proprium with respect to it, albeit just a generic proprium. It, accordingly must be predicated as a proprium, and that is  per se  predication. Further we have seen that what is signified by a proprium ontologically lies wholly within the carrier (it is the phenotypic expression of the Essence).

Similarly, the term  ' quality '  (now not following Ockham) is predicable universally and  per se primo modo  only of those terms in nominal definitions of qualitative states in which it (i.e. quality) is included. For example :  the nominal definition of the term  ' the red '  (which signifies a qualitative state) is :  something in which redness is present.  And indeed, the term  ' quality '  can be predicated  per se primo modo  of the term  ' redness '  (as in redness is a quality). This predication is possible because the definition of the term  ' redness '  is :    ' a color, such that ... (indicating its location in the rainbow) ... '.  And this definition ultimately leads to the term  ' quality '  as soon as we're going to complete it by ascending from  ' color '.  And if we do so, we will end up at the term  ' quality '.
But, as we saw with quantity, any infima species of quality, say greenness (or redness for that matter), cannot be universally predicated of any abstract qualitative term :  we cannot say :  ' redness is greenness ,'  nor  ' blueness is greenness ',  because greenness differs from redness and from blueness. Nor can we say :  ' quality is greenness ',  because not every quality is greenness.
An infima species of quality can only be predicated of a substance, and the term, expressing this species must be concrete but then only contingently so (if we dismiss Ockham's thesis that quality is exeptional among the accidental categories, and if we follow him as he did with quantity). An example would be :  Socrates is tanned.
And, like in the case of quantity, we must consider how things are when a certain quality, for example the blueness of  a  CuSO4 . 5H2O  (copper sulfate) crystal, is  intrinsic  to that (here a copper sulfate crystal) which has this color. Indeed, the blue color of  a  CuSO4 . 5H2O  crystal is an intrinsic feature of this crystal species, pervading the whole crystal. It is a generic proprium for that crystal species. And we have seen that a proprium always lies completely within the carrier, that is, it belongs to the Essence. The same was the case with the size of a given atom (considered above). Both must, therefore, be  per se  predicated of their subject, that is, predicated as propria.

So in the case of quantity as well as in the case of quality, there ocur two possible ways of predication (depending on what is signified by the subject term and by the predicate term), namely  per se  and  per accidens ( = contigent) predication. And this disrupts the uniform status of quantity as well as of that of quality. That is, it disrupts a fixed relationship between substance and quantity, and also between substance and quality. We now have two ways of :  quantitatively (or qualitatively) determined.
In fact, the above discussed feature of transcendentality cannot apply to both relationships at the same time and in the same respect, namely (1) the relation of a given quantity (or quality) with its substance, as it is expressed in  per se  predication, and (2) the relation of a given quantity (or quality) with its substance as it is expressed in a  per accidens  predication. As I said, these different relationships indicate two different ways of  quantitatively (or qualitatively) determined.
And this means that we cannot generally say that quantity (or quality) is transcendental, but only that things are intrinsically or extrinsically quantitatively (or qualitatively) determined.
A given crystal, being, say, 1 cm long, is  extrinsically  quantitatively determined with respect to length (It has grown to this length, and can grow further).
A given atom, being, say, 1 Angstrom in diameter, is  intrinsically  quantitatively determined with respect to diameter.
Quantity can behave differently in its relation to substance on different occasions. Therefore it cannot be a transcendental. We see this clearly when we compare  quantity  with  unity  (i.e. to be one), which latter is a genuine transcendental :  Every being whatsoever is intrinsically one, and always intrinsically one.
The same goes for quality. Also it behaves differently in its relation to substance at different occasions, and cannot therefore be a transcendental.

According to Ockham, quantity (quantitas) is not a thing different from substance (and from quality). It is not  a  res additum  ( = a thing added). Therefore we cannot say that a given substance (such as an atom, a molecule, a crystal, or an organism) has quantity. But we can also not say that a given substance  is  quantity. Neither can we say it in the sense that substance is a species of quantity (A species of quantity is, for instance, 1.70 m length), nor (can we say it) in the sense of a given substance being the same as quantity, or  a  quantity.
But a concrete quantitative term can contingently be predicated of a substantial term :  Socrates is ( = happens to be) 1.70 m long (while we cannot say :  Socrates is [ = happens to be] 1.70 m length, or is a length of 1.70 m).
The same goes for quality.
Among writers on Aristotelian metaphysics it hasn't occurred, it seems, that, as we have seen above, in some occasions certain quantitative terms, and also certain qualitative terms, when predicated of certain subjects, must be predicated  per se,  while on other occasions they can only be predicated  per accidens. The problem is confusing, and probably demands much more study, especially with respect to my assumption just made about writers on metaphysics.

So, to reiterate, a (concrete) connotative term, whose abstract counterpart signifies something belonging to one of the accidental predicaments, can, in some cases be  per se  predicable of a subject -- a  CuSO4 . 5H2O  (copper sulfate) crystal  is  blue -- meaning that this term, as a result of this predicational context, belongs to the Predicable  ' Proprium ',  while this same (connotative) term can, in other cases, only be predicated  per accidens  of a certain subject (as in :  my bruise is blue  [a bruise can also be red] ),  and is then, as a result of this predicational context, belonging to the Predicable  ' Accidens '.
So the  ontological status  of that what is signified by the abstract counterpart of the (concrete) connotative term depends on the predicational context (blueness in the one case belonging to the Essence of a copper sulfate crystal, and, in another case, not belonging to the Essence of a given being [my bruise, and, therefore, my body] ),  which (context) in turn depends on the meaning of the subject term.
If we have, say,  X Angstrom length  (of diameter), where X stands for a definite number of Angstrom units such that it indicates either, say, the size of such small things as a given atom or an atomic nucleus, or the size of, say, a given crystal, it depends on the meaning of the subject term of which  the concrete term  ' X Angstrom long '  is predicated, whether the significatum of the latter term's abstract counterpart -- ' X Angstrom length ' -- either resides totally within the carrier or substrate, when, namely, the subject term indeed refers to something like a given atom, or atomic nucleus,  or  whether it resides outside the carrier or substrate, when, namely, the subject term refers to something whose exact size is extrinsic to it, like a crystal  ( For very small things it will be hard to give an example of extrinsic size, because in that range of things  sizes  are usually intrinsic. For larger things an extrinsic size is easily given :  the size of a given crystal).
The same can be argued for quality.
So again :  Things can be quantitatively (or qualitatively) determined in two ways :  per se  or  per accidens.  And this makes the question as to whether  ' quantity '  (or  ' quality ,'  and also every one of the remaining accidental predicates)  is a transcendental or not, futile. Sometimes it is  a  res addita  (something added), and then it is not a transcendental, sometimes it is not, and then it is a transcendental. In fact this boils down to the finding that the accidental predicates are not transcendentals.

This concludes our general thematic framework, as a preparation for developing an intentional logic, in which (framework) we have discussed, more or less in the context of Ockham's philosphy, the ontological status of determinations, namely whether these are essential parts or replaceable determinations.

We will now discuss some details --again more or less in the context of Ockham's philosophy -- that we deem important and instructive for a development of an intentional logic, and especially for (further) vindicating a substance-accident metaphysics (against at least the strong version of nominalism), which metaphysics can serve as a basis for an intentional logic.

Natural Science and Metaphysics

Things like the relationship Substance--Accident, Matter--Form, and the like, seem to be assessed by Ockham as principles of natural science, because they are principles of material change. This seems to me correct. But what is not correct is saying that because they are principles of natural science they do not belong to metaphysics. The principles of natural science cannot themselves belong to natural science, because they cannot treated by or subjected to the very methods of natural science (observation, induction, deduction, verifying (or falsifying) observations), because these methods already presuppose these principles. The principles of natural science are therefore the  " metaphysische Anfangsgründe der Naturwissenschaft "  ( = metaphysical bases of natural science), that is, the metaphysical presuppositions from which natural science can get started.
And of course, there is no such thing as  ' principles of metaphysics ',  because there is nothing  ' below '  metaphysics, and if there were, it would belong to metaphysics.
So it is wrong to say that the book of Categories  " is not a book of metaphysics " (MOODY, p.144, note 1) because it is about substance and accident (which it is) and therefore about change.
Natural science is, it is true, about changeable things, such as atoms, molecules, crystals, animals, and stars. But it is not about changeable things only, and precisely, insofar as they are changeable things.
It is metaphysics which explains the very changeability-as-such of all material things :  And if we only recognize as real beings material beings, then metaphysics is about being qua changeable, which is equivalent to being qua being.
And not only natural science, but also logic (in the intentional sense), is based on metaphysics.

Some (further) notes about the ontological status of the accidental Categories.

There are two main species of quantity, viz., continuous quantity and discrete quantity. In connection with the ontological status of accidents, it is instructive to separately discuss them.

Continuous quantity
A continuum, as such representing continuous quantity, has parts that have common boundaries (therefore the parts are no more than potential parts, while in a discretum [ = contiguum] the parts are actual because they have no common boundaries).
So if we say that a given substance is continuous we can define the quantitative term  ' continuous '  as :  something in which (if internal boundaries are present at all) only common boundaries are present.  In a continuous body these boundaries are straight or curved surfaces. These surfaces are evidently not things that are distinct from the (continuous) substance (the body) itself and then added to that substance. They are not something other than a substance (or a quality, as Ockham adds). So continuous quantity, in the sense of something being continuous, does not mean that something is added to that something, making it continuous. We can only say that in making something continuous (which initially was just a contiguum) something has been removed, namely the interstices of the contiguum, resulting in it to become a continuum. However, a contiguum is not a genuine being (from which we then remove something), but a multitude of beings (namely its actual parts) to which these interstices do not belong.
Indeed, every genuine substance must be a continuum, that is, must have continuous quantity  per se, i.e. all by itself. And even if there are substances which are (nevertheless) contigua, they must have ultimate parts that are continuous, and they are continuous all by themselves. So continuous quantity is, as it seems, a transcendental (if we only accept as real beings material beings). Continuous quantity is necessarily implied by something being a substance. It is not added.
Line '  (whose significatum is a [one-dimensional] continuum) is a term signifying something as determinate with respect to length (or as limit of a surface). But length is not something distinct from that which is long, since any substance having parts with relative position to one another (assuming such partitioned things -- in many cases at least -- to be substances), can, merely through motion of these parts (rarefaction or condensation of that substance) become longer or shorter, without anything being added to it or subtracted from it. Thus  length   cannot be said to stand for something distinct from that which is long, but present in it -- just as redness does stand for something distinct from that which is red, but present in it (according to Ockham). So  ' length ',  and any other quantitative term, and generally the term  ' quantity ',  is not, according to Ockham, signifying a real accidental form after the mode of quality resulting in the different kinds of quantity being just so many species of quality. And this means that, still according to Ockham, these quantitative terms are transcendentals.
And dismissing the special status attributed by Ockham to quality, we say the same things about, say,  redness  (which is a quality), of which indeed  a  ' non-theological Ockham '  would say that it does not refer to something added to the thing which is said to be red. So also redness, and with it all other qualitative terms, are, according to non-theological Ockhamism, transcendentals.
Earlier, we have seen that in certain cases  length,  or, generally, size, is (indeed) a quantitative  per se  determination of the thing having this length or size.
In the same way a quality must in some cases be attributed  per se  to a given thing.
In other cases, however, they can only  contingently  being attributed to given things representing a species of substance. In such cases the (contingent) determination cannot, because it is contingent, necessarily follow from the Essence of the thing. It must be extrinsic, and therefore it is a distinct entity, distinct from the thing to which it contingently is attributed. It is a replaceable determination, residing as such outside the carrier (substrate) (of such determinations). Therefore it is added, and is thus not a transcendental.
But above we said that when some substance is either rarefied (for instance by heating it), which means that when it is a (physical) body it expands, or concentrated (for example by cooling it), which means that it then contracts, it does so  without anything being added.  Is length or size then transcendental after all?
I don't think so. The just mentioned  " without anything being added "  refers to physical material not being added (or removed). However, when we are speaking about transcendentals (such as UNITY, THING, and the like)  the  " without anything being added "  refers to the comprehension or formal content of the relevant concept :  Predicating a transcendental term of the term X does not add anything to the content of the concept of  X.  So the  " without anything being added "  in the case of rarefaction or concentration as changing the length or size of a given individual substance, is not relevant to the question as to whether length or size of an individual substance are transcendentals or not.

Discrete quantity
Numbers, in the sense of how many things there are (for instance in a given container), is a discrete quantity.
Since a (natural) number signifies a multitude of individuals simul sumptis (simultaneously taken), that is coniunctim (that is, the individuals taken in the manner of, ... and ... and ... ), and since any universal concept can determine such a multitude, a number, taken in itself, is, for Ockham, as for many modern mathematicians, a  " class of classes similar to a given class ".  For example, suppose that the given class is the set of regular polyhedra (tetrahedron, cube, octahedron, dodecahedron, icosahedron). Then we can point to many other classes or sets, for example :  these stars  * * * * *  (to which I point with my finger), these letters  A B C D E ,  these arms of a common starfish, etc.
All these classes or sets are similar to the first class -- the given class -- in virtue of the fact that we can relate these classes by a one-to-one correspondence of their elements. Signifying the elements by arbitrary symbols, the one-to-one correspondence looks like this :

And the whole set consisting of all these classes we call  " five " .
So with the term  ' five '  we refer to the total collection of classes that can be related to the given class by this one-to-one correspondence of their elements. So the number of elements of any one of these classes is determined by this correspondence. And it is we who have made this correspondence, that is, if ever we have added something to such a class, it is not something real. Also to its elements we haven't added anything. The class of objects remains precisely the same, whether we bring its elements into correspondence with those of other classes or not. So it having a definite number of elements does not add anything to it.  So  five,  expressing the number of elements of a class or set of them (considered as a single entity), is not a distinct entity added to that which is five-fold.
And, moreover, still thinking of  five-foldness  as an accident of a given substance, often such a collection of things (five things) is not a single substance. Only when  five  refers to parts they could be parts of a single substance, as in the case of the star-fish (consisting of five antimers [ = counterparts] ).  The parts themselves of such a star-fish are, however, not substances.
Now it is possible to hold a different view on these matters :
Ockham directs his criticisms as regards the status of accidental predicaments mainly to Duns Scotus. And that means that he criticizes the position that accidents are genuine things. But the view of St Thomas regarding these accidents is different from that of Scotus. It is such that we could interpret accidents not as things but as conditions or states of (genuine) things. In this way accidents are considered to be not beings in the full sense, but just auxiliary beings.
Let's see how this works out in the case of  discrete quantity.
If we have a collection of five things, then we could say of each of these things that they are in a condition of co-existence with four other such things  (the same view can be held, by the way, for  relation  [between things] ).  In the case of a number of parts of one given genuine substance the accident  fiveness  refers to each part :  In a common star-fish this fiveness or five-foldness is  per se  with respect to the (whole) star-fish, that is, it resides inside the carrier (substrate) and is therefore not a res additum (a thing added). But in the case of, say, five crystal individuals, present, say, in some natural or artificial container, it is just a condition or state, that is, it is a replaceable determination, and as such  a  res additum,  where  " res "  does not signify a full-fledged being (i.e. an ontologically complete being), but an auxiliary being, wholly in line with the equivocal nature of Being NOTE 10 ).  .
So of  ' discrete quantity '  one cannot, just like that, say (as Ockham does) that it is not a res additum. Sometimes it is.

Quantity of it self allows being equal or unequal to another quantity. In this way  3 = 3  and  4 > 3.  And  quantity of itself  could be equivalent to just a description of a given substance, implying that the equality or inequality do not apply  per se  to the individual substance, for no substance is of itself equal to another, nor is one substance equal to another by any form or thing really distinct from it -- rather it is equal to the other insofar as it is signifiable by a term connoting an operation of division (counting potential parts, that is, measuring length and the like), or of enumeration of parts (counting actual parts), applicable to the other (MOODY, p.155/6).

It perhaps sounds strange that two substances are equal (in length, say) (or unequal for that matter) to each other  "insofar as it [that is, one of them] is signifiable by a term ...".
That a substance is signifiable in a certain way must have a cause in extramental reality, that is, in the substance itself  and / or  in its environment.
Again, we agree that a substance to be equal (qua length, say) or unequal to another, is not a matter of some thing being added to that substance. What is added is not a thing but a condition or (equivalently) state.
A given substance  X  can happen to co-exist with another substance  Y  (existing somewhere) having the same length (where the particular length can be a proprium or just a replaceable determination). Then we can say  " X is equal to Y "  (with respect to length). Further we can then say that  " is equal to Y "  is a condition or state of  X,  albeit a very extrinsic and variable state (because this particular state of  X  already gets replaced by another as soon as the length of  Y  is replaced by another).
For substance  Y  we can say the same :  " is equal to  X "  (qua length) is a (very extrinsic) condition or state of substance  Y.

By Ockham  quality  is not considered to be on an equal ontological footing with  quantity :  While quantity is considered to be a transcendental, meaning that it has, apart from substance, no existence at all whatsoever, quality has  ' possible independent existence ',  that is to say, it is something that is carried, and, therefore, distinct from its carrier, implying that it could (by supernatural intervention) without involving contradiction, be separated from its carrier and subsist ontologically independently. This is equivalent to saying that quality is considered by Ockham to be a predicamental accident (but still in a stronger way (it to be) a being than St Thomas would consider it to be [and any other predicamental accident] ).  When we do not pay attention to this supposed supernatural possibility and its demand on the ontological status of quality, but follow the 'modern Ockham', then quality must be considered on an equal footing with quantity (with the exception perhaps that quantity stands nearer to Substance than quality does, as can be surmised from what has been found out about continuous quantity). And when quantity does not refer to a thing, so it is with quality.
A strong indication as to the ontological equivalence of quantity and quality is the fact that they cannot exist without each other :  Quality implies intensity, and intensity has a quantitative nature of its own, different from that of pure extension. So quality implies quantity. And quantity implies quality, that is, it turns out to imply (at least in our world) quality. If this were not so, all quantitative relations would not be observable, but we know they are observable.
So quantity and quality, although not being the same, are (ontologically) equivalent.

The 'modern Ockham' wants to consider quality, like quantity, to be not  a  res additum  (thing added), but just a description of the substance itself. But, as we saw with quantity, it is a determination that can occur in two ways :

So also here (that is in the case of quality) the ontological status is dependent on the predicational context (for example whether  ' blue '  is predicated of  ' copper sulfate crystal ',  or of  ' human body ' ),  which is in turn dependent (in addition to the meaning of the qualitative term) on the meaning of the subject term (as we just saw it in the examples).
In the case of some given quality being a replaceable determination of some given substrate it is  a  res additum,  but only in the sense of a condition or state of the given substance, which comes close to the position held by St Thomas with respect to the ontological status of accidents.

In the Category of  Quality  are included those terms (other than essential differentiae [essential quales] of substance, but see below), which constitute appropriate answers to questions concerning the character (quale) of things. As Aristotle says :  " Quality is a term that is used in many senses ",  and hence there is no single concept or meaning corresponding to the summum genus  ' quality '  that is univocally predicable of all four of the species of quality enumerated by Aristotle (Said a bit more precisely :  There is not one single intentio animae or natural sign corresponding to the one conventional sign, that is, the word  ' quality ',  but there are many (such intentions, concepts)).
However, it is probable that they all can be reduced to a single sense of quality, certainly if we limit ourselves to  qualities  that are ontologically relevant.
A qualitative term signifies habits or dispositions, habit differing from disposition in that it is with difficulty acquired or lost, while disposition is easily acquired or lost. Now what can be said to be acquired or lost by a substance must be something really distinct from the substance which is said to acquire it or to lose it (MOODY, following [his historical account of] Ockham, p.162).
If we interpret the  " really distinct from the substance "  not as being a genuine thing, distinct from any given substance, but as an accident in the sense of St Thomas, or better still, in the sense of a condition or state -- but involving only a formal content, and not a size, a number or any other quantitative feature -- of a substance, then we have interpreted  ' quality '  as a replaceable determination of some given substance. We know, however, that a quality (as such belonging in a Predicament), can sometimes be a phenotypic essential part of a substance (while the substantial differentiae, mentioned above ,  and (provisionally) excluded from the predicament quality, are genotypic essential quales), like  blue  is in the case of a copper sulfate crystal. In that case it is not just a condition or state of the substance, that is, it is not a replaceable determination but a permanent determination of it, and as such an essential part of it. So, like the substantial differentiae, such permanent determinations must be excluded from the predicament. But, of course this is not right :  No qualitative determination should be exclused from the Predicament  Quality .  The differences only appear in the different predicational contexts :
If the predication is  per se  and  in quid,  then the predicated qualitative term designates the genotypic Essence of the (given) substance, either completely or incompletely. Such a term is then called a species or a genus.
If the predication is still  per se  but now  in quale,  then the predicated qualitative term designates  either  a further determination of the genotypic Essence, and then such a term is called a differentia,  or  (the predicated qualitative term designates) a phenotypic expression of the (genotypic) Essence, and then such a term is called a proprium.
And, as we have said earlier, Ockham should have interpreted  Quantity  (and all other predicates belonging in all other Predicaments as well) in the same way.
Consequently (that is, continuing the just given characterization of  Quality  [which as such cannot be defined, but only circumscribed, because it is a supreme genus] ),  in the Predicament of Quality there are included all those qualities which Ockham calls "absolute" (having, according to Ockham, possible [ontological] independency, which substances actually possess), which (qualities) are signified by abstract terms (such as  ' redness ' )  that can stand disjunctively for either genotypical essential quids and quales, or phenotypical essential quales, or (replaceable) qualitative states, in extramental reality. Also the concrete counterparts (such as  ' red ' )  of these abstract terms are included in the Predicament of Quality.

An example of  a  relation  is that between  mover  and  moved.
Both the terms  ' mover '  and  ' moved '  are concrete (meaning respectively,  that which moves  and  that which is being moved) and are correlatives. Each of them can stand for individual substances.
But for what individual substance can the term  ' motion '  which is abstract, stand? It cannot stand for any individual substance taken alone, says Ockham, and it does not signify any "universal thing" distinct both from concepts in the mind and from individuals  in re  (that is, in extramental reality). Hence, if it is a term of first intention, it stands conjunctively for the two individuals that are signifiable disjunctively by the concrete terms  ' mover '  and  ' moved '  (MOODY, p.157). We could, of course, also hold that  motion  is a relation between mover and moved.
The significatum of the term  ' motion ',  like that of  ' paternity ',    ' similarity '  is, according to Ockham, not a res additum, that is, not a thing added to a given substance. However, I would say, that  ' motion ',  in the sense of  in motion,  is not just a description of some individual substance (here a physical body) without something added to it. It is  a  state  of a given individual substance (a state of motion), which as such is a replaceable determination.
One could counter this by saying that motion is always relative, and involves some other substance (physical body) with respect to which it moves. As such, motion allegedly is not a determination of this particular substance, because it could at the same time be at rest with respect to another substance (physical body). But even then, the given substance is in a -- albeit very extrinsic -- condition or state of motion relative to that other substance, and it is in a condition of rest relative to yet another substance. This complies well with what we said above, namely that motion is a relation (of a certain kind) :  The substance A possesses the relation  moving with respect to B,  and, in addition to it, the relation  to be at rest with respect to C. And moreover, it could be that motion is not relative after all, as is argued in the document about the  impetus  in Part XXIX Sequel-5 in Fourth Part of Website .  In that case motion is a true quality that a given substance can happen to have.
So the term  ' motion '  does signify a res additum, again (only) in the sense of a replaceable determination (not in the sense of a fully-fledged being or thing) of a given substance (physical body).
So generally, although we cannot say  "this substance is motion",  we can say  "this replaceable determination is motion".
Likewise we cannot say  "this substance is similarity", or  "this substance is paternity", but we can say  "this replaceable determination is (exactly specified) similarity", or  "this replaceable determination is paternity". In all these cases the replaceable determination is a particular state in which some given substance might find itself.
In this way it is perhaps correct to say that all relations can be reduced to a state of some particular single substance, and that means that  relation  is reducible to  quality  or  quantity.
There are two main types of relation, viz., predicamental relations (which are real relations between substances, for example the relation of friend between Socrates and Plato) and transcendental relations. The latter relations are such that both the one relatum and also the other directly constitutes the relation, such as, for instance, the relation between substance and accident (substance is a relation to accidents, accident is a relation to substance), the relation between matter and form (matter is a relation to form, form is a relation to matter), etc.
A predicamental  relation  is always just a condition of a given substance, that is, always a replaceable condition, state or determination, and never an essential part (proprium) of a single given substance. This is, because such a relation always points to another substance, and thus involves an extrinsic element. To say this more precisely :  a relation of a relatum always reaches beyond this relatum, namely to the other relatum). But such a condition  exists  really, it is not just an act of a mind comparing things.

The term  ' teacher '  also signifies a relation between at least two individual substances, here two human individuals (or a trainer and his pet). According to Ockham  ' teacher '  does not signify an entity in extramental reality distinct from the individual substances involved. It is therefore concluded by him that the only things which are  per se  in the Category of Relation are  terms -- namely, any term which cannot be verified of anything for which it can stand, without reference to some other term connected with it by a preposition or by an oblique case-ending NOTE 11 ),  this other term being called its correlative (MOODY, p.159, and referring to Ockham's Expositio Aurea, II, 64 r.).
The fact that a certain term can only be verified as to what it means by invoking another term (added to it by a preposition or by an oblique case-ending), must have a cause in extramental reality. And this we mean that a thing, a given individual substance, can find itself in a certain condition (state) by being in a certain environment, which either has resulted from its own actions (as in the case of 'being a father') or from the presence of other things which are independent of it, that is, happen to be present (as in the case of 'being similar').

The OTHER CATEGORIES (that is, other than Substance, Quantity, Quality, and Relation) will be discussed elsewhere, if necessary.

All Categories, except Substance and Quality, are, according to Ockham, modes of signifying individual substances or qualities either conjunctively or connotatively, and they signify nothing  per se  distinct from individual substances or qualities (MOODY, p.172).

Let us now, but still in a preliminary way, wrap up results (MOODY, p.172/5).
In all this we try to get as close to Ockham's undertaking as possible.

We can state two points of departure :

So if the different Categories signify all possible basic types of things, then of each such a type of thing we have its supreme genus. And although this supreme genus is not definable (because it does not have a higher genus), all the lower genera, down until the infima species, are in principle definable. For instance :

A mammal is a milk-producing vertebrate

(where 'vertebrate' is the genus, and 'milk-producing' is the difference).
Another example is :

red is the first-rainbow color

(where 'color' is the genus, and 'first-rainbow' is the difference).
This means that we can in principle know a thing completely, or at least signify it completely, namely by enumerating all the differences and all the (sub)genera up until the supreme genus (summum genus, genus generalissimum) of that thing.
However, the only things that exist are individuals. So we must completely signify such an individual thing as to what it is, and as to what it can become, where it is, etc. And this is impossible, because indeed individuality involves an almost infinity of relations, and, in the course of time, an almost infinity of features, and still more relations.
From this it is clear that the distinction of only some ten or less 'fundamental' Categories of things is not some sort of basic classification of individual things, because every single individual thing is as incompletely knowable as any other. We could just as well classify these individual things according to, say, their stereometric basic form, and then end up at only a few basic higher types, such as  ' Homaxonia '  (spherical forms),  ' Monaxonia '  (cylindric and conic forms),  and  ' Stauraxonia '  (pyramids) (See for such a classification actually carried out :  the Series of documents under the main heading  BASIC  FORMS  in Second Part of Website ).

From all this it is evident that the Categories of Aristotle do not mean to express different types of individual things (as Duns Scotus would probably have it), but are intended to give all the basic ways of signifying one and the same individual thing.
Of course we could, alternatively, try out the following possibility :

While the first category signifies (in extramental reality) individual things, that is, genuine beings, or at least their essential parts, the other categories (the accidental categories) signify (in extramental reality) beings that are only beings in a certain respect, which here means :  just auxiliary beings, having not the full being of genuine beings. So while the term  ' animal '  signifies individual beings (disjunctively), the term  ' redness '  signifies an auxiliary being.
However, this alternative view does not comply with the first thesis of Ockhamism, namely that only individuals exist (in extramental reality), because, say,  ' redness '  is not in virtue of itself, that is, not  per se,  i n d i v i d u a l  ( It is individual only in virtue of the individual substrate by which it is carried). So the individuality  of  redness  (that is the significatum of the term  ' redness ' )  is not a feature of it, and thus  redness  as such is not individual. And this, consequently, means that  redness  as such does not exist in extramental reality, which in turn means that we cannot say that the accidental categories signify  'somethings' in extramental reality. So indeed we are left with the accidental categories being just ways of signification of ----- any given substance, which itself is a way of signifying a thing, namely substantially (other ways are :  quantitatively, qualitatively, relationally, etc.).

With this interpretation of the Aristotelian Categories we in fact express the fact that we do not know all things, and also we express the fact that any given thing can be signified in diverse ways, in turn expressing that we cannot, with one term signify a thing totally (that is, comprehensively, completely) as it is or can be. We can only signify it according to one or another category.

In the light of the above considerations we are now able to follow, paraphrase, and comment on what MOODY, p.172--175 says when summarizing the results of his investigation about in what way Ockham interprets the Categories (Predicaments) of Aristotle :

Ockhams treatment of the Categories of Aristotle is guided by two concerns :
First he seeks to preserve the distinctions between the categories [among themselves] as forms or modes of signification, from the combination of which [forms of signification] affirmative and negative statements [predications, propositions] arise, though [Ockham] denying that terms are placed in distinct categories [just] because they signify really distinct things.
Secondly, he seeks to preserve the distinction between terms [signs] and the individual things signifiable by them, by insisting on the fact that  "being"  is predicated equivocally  [thus not that  "being"  is  equivocal, in the sense that there are different degrees of being ]  of the terms in the categories  [ We can say  ' a crystal is a being ',  or (we can say)  ' blue or blueness is a being '  (equivocal predication), where we, however, refer to the same being (univocal being), here, for example, an individual copper sulfate crystal ],  so that no term of first intention, when taken alone, signifies  that which is  by its complete and actual being, whereas every term in a category signifies  that which is,  in one of its senses, by abstracting from the signification of it in another of its senses. In other words, Ockham's whole aim is to preserve the relativity of knowledge to being, without destroying the distinction between knowledge and what is known  [ that is, between a being-insofar-as-we-know-it, and that being, how it is in itself independently of our knowing it ].
Ockham's effort to show that none of the terms in the categories signifies things that are really distinct from substances or qualities, preserves the possibility of signifying the same individuals in diverse ways -- i.e. it preserves the possibility of true affirmative predication  [ for example,  ' this white thing is learned ',  where we in fact say that the given thing as we signify it by the term  ' white '  may also be signified by the term  ' learned ' ].  For [as MOODY continues] if the accidental categories signified distinct things  in re  [ = extramental reality] [as the term  ' quality '  does in Ockhams view], [then] there would be no difference between them [accidental terms] and qualitative terms with respect to mode of signification  [ And if we treat quality as ontologically equivalent to the other accidental categories, then we must say that if the accidental categories signify distinct things  in re,  they all must signify in the same way as the term  ' substance '  does ],  and thus the different genera of accidental categories would differ only as one qualitative term differs from another, namely through indicating a different quality. In that case there would be no more reason for putting  "quantity"  and  "relation"  in different categories, than for putting  "color"  and  "heat"  in different categories. Or if  "quantity"  and  "relation"  constitute distinct categories because they signify distinct things, then coordinate species in the category of substance, or in that of quality, should constitute distinct categories by reason of the fact that they signify really distinct things.

[ If  ' quantity '  refers to a thing (and quality to another), it refers, among others, to, say, a given length, but signifying it in a maximally indeterminate way. If this length is considered to be a thing, then a given dog (incompletely signified by  ' substance ' )  is another thing, but then a given cat is yet another thing, meaning that they should be placed in different categories. ].

[ So while Ockham says that when all accidental categories, except quality, were to signify distinct types of things (as, for instance Thomism holds), they must signify in the way quality does (because all these non-quality accidents on the one hand, and quality on the other, are in this sense the same that they are not subsistent), we must say (by denying Ockham's thesis that quality is -- although not subsistent -- a thing) that when all accidental categories were to signify distinct types of things, they would signify them in the way the term  ' substance '  does, because when they do not, they would, compared to what and how the term  ' substance '  signifies, signify  different  types of things in  different  ways. To say it again :  A term of a given accidental category would signify a thing that is of a different type from that of what is signified by the term  ' substance '  and also (of a different type) from what is signified by the other accidental terms, and moreover, it signifies (that thing) in a different way, different from the way the term  ' substance '  does it (i.e. the term  ' substance '  signifying a {fundamentally different} thing), and also (the mentioned accidental term signifies) in a different way, different from the way by which the other accidental terms do it (i.e. signify {fundamentally different} things).
So we see that if we let accidental terms signify distinct types of things  in re,  and if they do not signify them in the same way as the term  ' substance '  does, then we obtain total indeterminacy (namely the mentioned :  signifying fundamentally different things in {moreover} different ways). Only when, while (letting the accidental categories) signifying things, they do it in the same way as the term  ' substance '  does, there is determinacy, because then we have just a classification of different types of things.  But then we have left Ockhamism, because Ockham holds that the categories list the different ways of signification.
And the Categories being a system of terms signifying fundamentally different kinds or types of things (beings) is precisely the position of Thomism.
However, we must realize that the discussion just had, does not demonstrate that when the terms of the categories are held to signify things, the implication would then be that we end up with Thomism, and not with Ockhamism :  Although, namely, the terms of the Categories, which are terms of first intention, do signify things (which is also conceded by Ockhamism), this does not necessarily imply that the system of Categories is then automatically a system of fundamentally different types of things not reducible to one another :  The system can still be a system of fundamentally different  ways of signification.  If it is both, then we have indeterminacy. But if the terms of the Categories just signify things, which they do, but do not thereby constitute a system of fundamentally different types of things, then we have only the different ways of signification. And what are the significata of these different significations? They are not correspondingly different things, because then the indeterminacy returns. What then are these things-signified? Well, they can only be one and the same thing :  Each extramental individual thing can be signified in several ways, and this is what the system of Categories is about. It is a system of the possible different ways of signifying one given single individual thing.  And this is precisely the position of Ockhamism.]

In short [as MOODY continues], the effort, so much criticized by Ockham, to interpret the categories as a division of terms according to distinctions between entities signified by them, instead of according to differences in mode of signification, can only result, when carried out consistently, in the destruction of the very idea of the categories.

It was said that the categories are about the differences in mode of signification (of individual things).
We could say that when we, as it has turned out, are forced to express an individual thing in different ways, or when we can (as it turns out) express an individual thing in different ways, this must have a cause.
Where and what is this cause?
It could be in the things themselves, namely their patterned ontological make-up. This is the position of Classical Metaphysics, which was followed largely on the present website.
But, as an alternative view, the mentioned cause could also lie in us, that is, in the knower. The different ways of signification express our incomplete knowledge of things. And this is the position of (a revised) Ockhamism, and it is worthwile to discuss it in the sense of an interesting alternative of Classical Metaphysics.

With respect to what was said at the end of the above paragraph, we can even go further :
Ockham held that Quality is not only different from the other accidental categories with respect to formal content, but also ontologically different.
All accidental categories were seen by him as not referring to something which is in a stronger or lesser degree outside the carrier, except quality. The only reasons were -- as far as I know -- theological in nature, having to do with the transsubstantiation dogma. This is, of course, for us unacceptable. But are there perhaps genuine philosophical reasons to consider quality to be ontologically different from the other accidental categories?
In Thomism it is argued that there is an order of nearness or proximity of accidents (of a given thing) with respect to substance (being the substrate of these accidents). Quantity is very close to prime matter, in virtue of continuity and divisibility (both are possible aspects of quantity). Being close to prime matter first of all means that quantity, especially continuous quantity, has substrate character. So while quantity immediately rests on substance (and mediately on prime matter), it is itself the substrate for all other accidental determinations, especially for quality.
Now it can probably be argued that  relation  can be reduced to either one of its fundaments, that is to either one of its relata (which are those entities that stand in the given relation to each other), which means that many relations can be reduced to quantity, and the rest to quality. It probably can also be argued that all remaining accidental categories can be reduced to quality (and maybe some to quantity).
So what we now have is, first of all, the  carrier  (substrate), consisting of prime matter and substantial form (Ockham denies prime matter to have ontological significance, but prime matter is necessary to explain the possibility of substantial change). This carrier (substance) carries quantity, and quantity in its turn carries quality.
If this is correct, we can say that quality is farthest away from substance. It is this feature which lets it ontologically differ from quantity, while still differing from substance in that it is ontologically dependent (on a substrate), while substance is not.
Now the philosophy of Ockham will have it that quantity is totally embedded in substance, turning it into an essential, although varying, aspect of any given substance, allowing the latter to be directly described, albeit incompletely, by a quantitative term. This, in contrast to quality, by reason of the different ontological nature of the latter.
A quality when signified by an abstract term (such as  ' redness ' )  can stand disjunctively for individual things in extramental reality, that is, these individual things together form the term's extension. That the abstract term can stand for these individual things does not mean that these individual things constitute the meaning of the term. They constitute the meaning, not of the abstract term, but of its concrete counterpart (such as the term  ' red ' ).  Both the concrete and abstract forms of such terms are included in the Category of Quality. But whereas the concrete form signifies a substance  by  connoting a particular quality really present in, and distinct from the substance, the abstract form (which appears in the nominal definition of the concrete term) signifies and denotes precisely the particular quality without connoting or indicating its presence in a substance (i.e. its being carried by a substrate).

However, despite this possibility to give -- with Ockham -- to Quality this special status with respect to the other accidental categories (while still differing ontologically from Substance), it is perhaps better not to assess quality that way, especially because we saw that quality lets itself be treated on an equal footing with quantity perfectly well. So for the time being we will consider quality, like quantity, as just another way to signify the individual thing.
This latter consideration will probably imply that the difference between  per se  and  per accidens  with respect to the attribution of quantity or of formal content (quality) to a given individual thing, does not imply a corresponding ontological difference. It probably boils down to be just a form of preliminary description of the thing as part of the overall description of it.

On the fact [MOODY, p.173, continues] that [the term]  "being"  is predicated equivocally [here, because of our factual incomplete knowledge] of the terms that are in the categories, depends the possibility of distinguishing between signs and things signified  [this is because  the term  ' being '  being predicated equivocally of the terms in the categories  means that a same individual thing can be signified in a number of different ways, that is, that same thing can be referred to by several signs.],  and also the possibility of adequating signs to that which is, through true propositions [that is, predicating these signs of terms that stand for the relevant individuals, which latter constitute the that which is].
The distinction between the categories is a distinction between different forms by which finite causes of change can be understood or apprehended. Hence, while every term in the categories is a form by which individuals can be apprehended, no single term signifies any individual by all that is. That is to say, every term in the categories abstracts from something  [ I understand this to mean :  isolates something ]  by which the individuals for which the term can stand exist actually under particular conditions  [ For example the term  ' walking '  is the result of an isolation of a condition or state something -- let us call it X -- is in, and X is then referred to by means of mentioning this state.  So although X is signified or intended -- by the term  ' walking '  -- and thus known, it is still as such very incompletely known.].  Every term in the categories stands either for substances or for qualities, either disjunctively, conjunctively, or connotatively. And hence every such term abstracts either from qualities (some or all) by which a substance can be apprehended to exist [can be detected to exist], or from substances by which a quality can be understood to exist.
In short, no term in the categories is an adequate sign of the existence of any particular individual, and yet it is not the sign of any existence other than that of individuals -- for "to be" is to be individual.
From this we can see how terms combined in propositions, either with the verb  "is",  or its equivalent, added to them, are alone adequate signs of individual existing things. We can see why no single term, and no definition, is either true or false, and why a term is said to be a sign  capable  of standing for individuals, but does not by itself stand for any individuals. Which is perhaps the reason why every term in the categories is of determinate significative [ = standing for what it means, and not for what kind of sign it is] character only  qua  universal, and the reason, also why no term of first intention can stand for anything other than individual things.
As a form of signification, every term that is  per se  in a category is universal because it is potentially a factor in the knowledge (through propositions) of individuals really distinct and different from each other [one sign for many individual things]. But precisely because one individual actually existent, being per se distinct in its existence from other individuals, cannot  of itself  be a factor in knowledge of other individuals, it follows that nothing for which any term of first intention  can stand,  is universal.
A consequence of this analysis is that the only things that can be said to be  "individuated"  are universal terms. The  "principle of individuation"  is the verb or copula consignifying time, which, when applied to a term that stands only potentially for individuals, forms a complex sign (proposition) which affirms that something is or is not. The very word  "individuate"  indicates the character of predication -- it is the act of  "un-dividing" [ = the bringing together {of terms}, resulting in an in-dividuum], an act applicable only to what has [initially] been divided [namely terms that are only there in an isolated fashion, and thus are divided], and which therefore presupposes an act of dividing [the] elements that exist together (i.e. "con-tingently") [elements, initially held by some act separate from each other, and therefore existing just contingently, are finally brought together, they are made un-divided, resulting in integration, that is, (resulting) in an actually existing individual]. All finite terms of first intention are elements abstracted either directly, or through the parts of their definitions, from experience of contingent or undivided existence. Until this abstraction is offset by an affirmation or negation which re-introduces the indication of  "being together"  in time, supplied by the verb  "is"  taken as of present tense, a term is not adequated to that which actually is, and hence is not true or false of actual existence. The abstraction is offset in another manner by propositions  de possibili, by which [the] terms are formed into adequate signs of  that which can be,  as is the case with univeral propositions.
But every term, taken alone, is only a form by which individuals can, in an act of affirmation or negation, be indicated or apprehended as actually or potentially existent. It is not itself an indication or an apprehension of the  existence  of individuals.
That terms can be analyzed and distinguished as forms by which individual being or nature can be recognized and signified, independently of affirmations of existence involving these terms, is what makes the analysis of terms given in the Categories a "formal" analysis. Though the analysis presupposes the existence of propositions (since terms are significant only qua potential elements of propositions), it does not presuppose knowledge of the truth or falsity of any propositions about things which the terms in the categories signify [recall, that although the terms of the categories do signify things or beings, this does not necessarily entail that the Categories are a system of types of things or beings. It can be (and is according to Ockham) a system of ways of signification of a given thing]. Hence it [i.e. the analysis of terms as classified in the Categories] provides us with a means of defining terms independently of questions of truth or falsity, and with a means of understanding what is affirmed or denied by a proposition independently of the determination of its truth or falsity NOTE 12 ).
The possibility of evident and determinate knowledge of the truth or falsity of a universal proposition, on which depends the possibility of demonstrative science, depends on the possibility of defining terms independently of prior knowledge of the truth or falsity of universal propositions in which such terms are involved.
Ockham's underlying concern, throughout his analysis of the Categories, is to show how this can be done, and thereby to preserve Aristotle's foundation for a science of nature distinct from a discursive or dialectical metaphysics [MOODY, p.175].

So far about MOODY's results concerning the Aristotelian categories as interpreted by Ockham.
To summarize :  According to Ockham the system of Categories is not a system of fundamentally different types of thing or being (not reducible to one another), but a system of different types or ways of signifying a given individual thing or being. But, although not being a system of non-reducible types of thing or being, its terms nevertheless signify things or beings, for instance :

The terms  ' man ',  ' animal ',  ' crystal ',  etc. -- all (taken) from the Category of Substance -- signify individual things or beings, by way of pointing to what they essentially are.
The terms  ' 1.70 m long ',  ' five-fold ',  etc. -- all (taken) from the Category of Quantity -- signify individual things or beings, by way of pointing to what they quantitatively are.
The terms  ' red ',  ' green ',  ' hot ',  etc. -- all (taken) from the Category of Quality -- signify individual things or beings, by way of pointing to what they qualitatively are.
The terms  ' teacher of ',  ' father of ',  ' congruent to ',  etc. -- all (taken) from the Category of Relation -- signify individual things or beings, by way of pointing to what they relationally are.

This position, as regards the Categories, is sharply contrasted against the views of Duns Scotus, but, although differing in some respects, is not so much different from the Thomistic position. For we can ask :  Why can we signify a single individual being in a number of ways? This must somehow be based in such a thing itself. How? Because such a thing or being is in fact an ontological complex consisting of a substrate that carries (inheres) several different types of auxiliary entities. We can then signify this complex by pointing to the substrate, or pointing to one of these auxiliary entities inhering in that substrate.

Terms and their meanings

In what follows we will dig deeper into the problem of what in extramental reality is in fact intended or meant by terms (concepts), either as taken alone, or when figuring in some context, especially in a propositional context.
The following list presents some properties of terms.

As to Signification and Designation (significatio and Suppositio) of terms :

With all this in mind we can discuss several problems about what in extramental reality is intended by certain terms.
Ockham's answer to the problem of (the ontological status of) univerals ( = universal terms such as  ' dog ' )  is :  Universals are terms (termini concepti) which signify individual things, and which stand for them in propositions. Only individual things or beings exist. And by the very fact that a thing exists it is individual. There are not and cannot be existent universals (that is, there cannot exist universals in extramental reality). There is not some one existing universal nature in the many individuals of a given species, because when one such an individual is annihilated, the others remain.
Of course there must be a reason (cause) that we can speak here of  "the individuals of a given species". The reason cannot be that each individual is identical to every other, because they are not. But this reason can also not be the presence of one (that is, numerically one) nature. The reason could be that a certain nature is exactly repeated over a multitude of individuals, as is supposed to be the case by Classical Substance-Accident Metaphysics, and which returns in our times as a basic presupposition of natural science :  Every (material) being has a Species-Individuum Structure. Natural science presupposes that there are  natures,  not subsisting by themselves, but being repeated exactly in several individuals (individual cases). We have spoken much about this, and stated that natural science is only possible when it presupposes the possibility of repeatable natures. But we have also said that if this repetition were not exact, but still very close to it, natural science can still exist and using its fundamental methods as long as the inexactness is below a certain limit NOTE 13 ).
Not  exactly repeating 'natures'  is equivalent to  similar thing  (say, a number of things, more similar to a given thing, than other things are).
The existence of the universal consists in an act of the understanding and it exists only as such. It owes its existence simply to the intellect :  There is no universal reality corresponding to the concept (as universal term). It is not, however, a fiction in the sense that it does not stand for anything real :  The universal stands for individual real things, though it does not stand for any universal thing. It is, in short, a way of conceiving or knowing individual things.
The universal concept arises simply because there are varying degrees of similarity between individual things.
On the basis of this similarity the specific and generic concepts emerge.
Socrates and Plato are more similar to one another than either is to an ass. And this fact of experience is reflected in the formation of the specific concept of man (its generic concept is formed by comparing man with (other) animals).
But we have to be careful of our way of speaking. We ought not to say that Plato and Socrates agree (share) in something or in some things, but that they agree (are alike)  by  some things, that is, by themselves, and that Socrates agrees with (convenit cum) Plato, not  in  something, but  by  something, namely himself (Ockham, In I Sententiarum, 2, 6, EE ).  In other words, there is no nature common to Socrates and Plato, in which they come together or share or agree. But the nature which is Socrates and the nature which is Plato are alike.
However, according to me, it is not easy to say what  "similarity"  actually should mean. We are inclined to say that two things are similar (but not formally identical) when there is something definite that is shared by both, while there are some other minor features which they do not share. But this is not the meaning (of similarity) as was stated above (no common nature for Socrates and Plato). But what then is it? Well, when two things are similar (but not formally identical) we usually describe it by means of these two things both possessing something formally identical, in addition to features not shared by them. So this is just a way of describing or clarifying similarity. What is it then really, when we want to deny that they have something in common (that is, that there is a definite something that is exactly repeated [resulting in two such somethings] ),  in these two things that makes them similar?
It is really hard to get rid of this exact repetition of something, and if we say that the repetition is not exact, meaning that that something is just similar, we're back at the same problem.
We can do it, perhaps, by using the example of the temperature of a material thing.
Two temperatures can be similar (but not identical) when they are close together, say, 120 C  and  130 C,  as the respective temperatures of two material things, (close together) as compared with, say, 120 C  and  40000 C.  Here we can indeed not say that the two things having respectively the temperatures of twelve and thirteen degrees Celcius have similar temperatures by the (alleged) fact that they both have, say, the temperature 12.50 C  in common, while the first object does not possess  +0.5 degrees while the other does possess it. The two temperatures are simply similar, or, said a bit differently :  The two objects have similar (though not identical) temperatures. And this type of similarity does not imply a catallel ( = dual, or two-fold) structure of the entities involved.
We can come up with still more examples of  similarity  not involving a catallel structure :
Crystal species can be characterized by Space Group (symmetry) PLUS Chemical Composition.
But when we look carefully to any two crystal individuals of allegedly the same species, then we see that in neither of the two individuals the characterizing space group symmetry is totally realized. Every crystal individual shows 'defects', that is, irregularities in its structure, destroying the alleged space group symmetry. And in every crystal individual these defects are different. So the crystal individuals are similar at best with respect to their space group symmetry. But this does not entail that they share some symmetry (other than asymmetry) and differ in other aspects of symmetry. Their symmetry is merely similar. The same goes for chemical composition as a result of impurities.
And, generally, if we have to do with extensive and complicated structures, they could be similar. However, they could be so (similar) without involving a catallel structure ( = one structural pattern shared, another not shared), as the following diagram suggests :

Figure above :  Diagram illustrating  similarity  without involving a catallel structure.
The nine columns are similar to each other, but have nevertheless no segment or series of segments in common. Each column shows ten different features. Only features at the same level (at the same height) are comparable at all. There are no two or more columns that are equal qua featural content.
If we go along the row of columns like a movie, we see the column change. And this is, it is true, 'creation in a thing', but without revealing a definite substrate of change, that is, without there being a common part, remaining the same during the whole sequence of change :  In the first two columns (counted from the left) we see a common part, namely the part consisting of the first eight features (segments) counted from below. But if we include the third column, this eight-segmented part no longer is common between the three columns, but only the seven-segmented part. If we include the fourth column, this is no longer a common part, etc.

All this does not mean that all similarities are like that. For example, if we consider the individuals of a given atomic species while they are in the same energy state, physical theory tells us that they are exactly (formally) identical to each other. Also when we consider two atoms of the same atomic species, but one of them being ionized (that is, having lost or gained one or more electrons), they are not (formally) identical anymore but just similar. But in this case we can with good confidence assume that their atomic nuclei are still (formally) identical, while their electronic periphery differs.
So what is important is, that a description of similar entities in terms of one part being shared (with respect to formal content) and another not shared, should not, without qualification, that is, without further evidence, imply a structure (that was described) to be itself consisting of a shared a non-shared part or aspect. It could be, but does not need to be so structured.
In the case of complicated things such a catallel structure is expected to be be rare (because the complicated structure consisting of many parts has a higher chance of one or more parts being different from the corresponding parts in the other individuals, destroying an otherwise shared super-part). Of course there will be small parts that are formally identical to the corresponding small parts in all the other individuals, but such small parts cannot reasonable be interpreted as a common substrate for change.
All this means that a description of similar things by means of shared and non-shared parts or aspects is often not more than  a  model.  And indeed in natural science, whose objects of study generally are messy indeed, one almost exclusively works with models. And we know now that such models, useful as they are, will not necessarily reflect a corresponding (ontological) structure objectively existing in extramental reality. A good example is the mechanical particle model of the atom. This model, displaying an atom as some sort of miniature planetary system (but with electrodynamic instead of gravitational forces at work), is still useful in many contexts. However, in many other contexts it doesn't work anymore, because of the wave-like properties exhibited by the 'particles' (constituting the atom) under certain conditions.
So the message of Ockhamism is that we should not, without qualification, and without further evidence, transpose a way of knowing to a way of being. A catallel structure of the description (shared, non-shared) does not necessarily mean that the thing described is itself also catallel. In particular we must be careful not, without further evidence, transpose a way of predication or the structure of the definition,  to  a way or structure of Being.

So, Socrates and Plato (and Peter, etc.) are similar (they are not identical and at least differ already with respect to extrinsic features), and this similarity might be the result of having some definite entity in common (in the sense of such an entity exactly being repeated), but it need not be so.
And of course this destroys the universality of the idea of an Essence in things, that is the idea that in each individual thing of the same species an Essence resides in it which is formally identical in each individual. Seen in this way (which seems perfectly plausible until now) we cannot adhere anymore to the traditional Substance-Accident Metaphysics, insofar as this includes the existence of Essences. And without those Essences, the distinction between the  per se  and the  per accidens  with respect to a given species of thing disappears. The species (and also the genus) does not have any ontological significance anymore, because it is more or less arbitrarily formed by us on the basis of similarity of the relevant things.

In First Part of Website we established that a genuine substance (in the metaphysical sense) is in fact (the product of) a (totality-generating) dynamical system. The dynamical law of such a given system was considered to be the Essence of the thing (the Totality) generated by the system. And all things involving this same dynamical law are then individuals of the same species. But here again, the dynamical laws will never be exactly identical, but differ more or less. They are just similar. But again, this similarity could be the result of the shared non-shared duality, but not necessarily, that is, not in all cases. So also here the universality of catallel structure is destroyed. In many cases such a structure is no more than a model.

The view of similarity, as given above, has consequences for the ontological interpretation of  change.  Change of something, let us call it Xn ,  can result in something, Xn+1 ,  that is more or less dissimilar. That is, Xn+1 is not identical to Xn (because that is no change at all), but different :  Xn+1 can be similar (but not identical) to Xn , or more or less strongly dissimilar to Xn , or totally different from Xn .  This difference, be it little or great, can be described by :  One component staying the same, while another component is replaced by another. In this way we describe what is called 'accidental change' in Substance-Accident Metaphysics. It is in fact a metaphysical (ontological) explanation of accidental change :  Every being is (in ordere to explain this change) ontologically composed of two components :  a 'substance' (a penultimate substrate) and a set of inhering, but replaceable, formal contents.
In the case of total change (substantial change), on the otherhand, we must explain this possibility (as such conceded) by positing a second duality (catallel structure), namely :

In other words, to explain metaphysically, the possibility of total change, we must view the penultimate substrate in turn as being catallel (now consisting of the ultimate substrate and the substantial form). If we would not do this, then  total change  is not adequately described, because it cannot then be distinguished from :  total annihilation of something followed by the creation, out of nothing, of another something.
This explanation of change is in fact an ontologization of forms of discourse :  The (grammatical and logical) subject is identified with a substrate, while the (grammatical and logical) predicate is identified with some formal content, that either follows directly upon the ultimate substrate (prime matter) and is called the substantial form, or follows upon the penultimate substrate (substance) and is called the accidental form. The respective predications can be exemplified by  "this thing is a man"  and  "Socrates is tanned".

But we have found out above that the shared-nonshared structure is not universally valid :  There will be many cases where the difference between Xn+1 and Xn ('accidental' change) or between X and Y ('substantial' change) cannot be described by the shared-nonshared duality (if we nevertheless do so, then we know that it is just a model). So we cannot universally say that a being necessarily has a catallel structure, be it as prime matter--substantial form, or as substance--accident.
Indeed, we must not ontologically distinguish between the ability of a being to change and that being itself.  ' Having prime matter '  is just a description of a being with respect to its ability to change totally. While  ' substance '  (as substrate) is just a description of a being with respect to its ability to change but not necessarily totally so. They are just descriptions, because as we saw, the metaphysical explanation is not universally valid, and so does not say anything about the ontological constitution of a being insofar as it is a being. The 'metaphysical' explanation is just  a  model  of change, and nothing more. The model can be, and is, useful, but its structure is not necessarily isomorphic with some corresponding structure in extramental reality.

So we have in extramental reality :  individual things, that are more or less similar to each other, allowing a typological classification of them (genera, species). Such an individual thing has no Essence, residing in it, and necessarily be exactly repeatable (and repeated). And because such things do not have such an Essence, they do not have an intrinsic identity. They are interwoven with everything else, that is, all individual things are more or less strongly interacting with each other. So the world (the Universe) is one single dynamical system of interacting individual things. There is only one genuine Essence, and that is the Universe itself. And because this Essence is the whole Universe, there is no  per accidens  with respect to the Universe. All features are  per se  with respect to it.
Indeed, when we consider all consequences and implications of (amended) Ockhamism, we obtain a totally different world-view from that of substance-accident metaphysics. All the ontological distinctions made in the latter are degraded into no more than models, and it is expected that some of these distinctions have no meaning at all.
As a result of all this, the world becomes much more uncertain. Today's science works with models and therefore with approximations. The in itself more or less chaotical natural world of things and events is being idealized in order to get some grip as to its structure. Science is forced to isolate things and events in order to obtain some understanding, but it knows quite well that things and events are not in fact isolated.

All this seems to be a devastating critique levelled against the substance-accident metaphysics. Let us think about this.
Natural science investigates processes, and therefore change. It does not investigate change qua change, because that falls beyond its proper province delimited by its methods and presuppositions. Natural science investigates things, as to what they are. It does not investigate things qua things, neither does it investigate the what qua what, because of the same reasons. It investigates natural laws, but it does not investigate natural laws qua natural laws. And as has been said, in investigating all these processes, things, natural laws, etc. it is forced to use models. And it knows that these models are at best only approximations or idealizations with respect to matters in the real world.
On the other hand, it is all these  'x qua x's  that are investigated by metaphysics :  change qua change, thing qua thing, a natural law qua natural law, etc. In other words, it asks itself :  What is change? What is a thing? What is a natural law? etc. And now we see, that like it is in the case of natural science, metaphysics is expected to be forced to use models too. And indeed, on this Website (consisting, until now of five large main Parts) we have presented several possible metaphysics, such as the ontology of Nicolai Hartmann, the theory of  If-Then constants of Woltereck, a theory of holism, inspired by Bohm, Sheldrake and Meyer-Abich, and of course the substance-accident metaphysics inspired by Aristotle and St Thomas Aquinas.
The fact that one can reasonably present several alternative metaphysics points to the uncertainty of knowledge in this respect. And that's why all these metaphysics are not more than mere models. They are models of the structure of beings (only) insofar as they are beings, or of the structure of Being (only) insofar as it is Being.
And when we consider these models one by one we come to the conclusion that the  substance-accident metaphysics  (as is worked out extensively on this website [especially in First Part of Website] is the most promising model of all.
So we will hold that things are catallel, and that they are so two-fold :  prime matter--substantial form, and substance--accident (or dynamical law--system state). But what we have learned from the above discussion is that such a metaphysics is only a model of the ontological composition of beings.

Determinations and their ontological status.
Let us recap some results of the earlier discussions.  All accidents are signified by connotative terms (such is a term referring to one aspect primarily and to another aspect secondarily). And this means that they all signify substance, that is, the individual thing. But they signify it by a detour. The detour is connoted. This  signifying by a detour  is just a way of describing that individual thing, and this way should not, without further argument, be ontologized. Thus the description  Socrates is 1.70 m long  should not, without further argument, be interpreted as referring to a real thing consisting of two formally distinct parts, a carrier and a carried (recall our above discussion about the shared-nonshared duality). It is just a way or model to describe the individual Socrates.
If we ask  "how long?",  we do not mean (to ask) how long this quantity is, but we ask how long this substance is. The only things that exist are substances, and therefore 1.70 m length is not a thing (added to a given substance).

Within our revision of classical substance-accident metaphysics, as has been carried out in First Part of Website, the concept of condition or state was of great help. Now we must eliminate it again, if we want to investigate along Ockhamistic lines :  A state is a state of something. In fact it is a natural stadium, reached now, of a dynamical system, or it is the result of a 'perturbation' that has affected the system from 'outside'. But this latter (i.e. the influence from outside) involves some (other) dynamical system that has its 'own' states. Here we see again that we, in order to explain things, isolate something, let us call it X, and interpret observed features always in relation to X.
But we now know that this is just an idealization or model :  A given crystal is 1.70 cm long. Here, something is isolated which is then called 'crystal', for example a snow crystal, which finds itself in a certain condition or state which is here very incompletely indicated by '1.70 cm long'. The latter is the result of the stadium now being present, the particular stadium of the dynamical system CRYSTAL, and of factors being at work from 'outside' this system. A next stadium and new alien factors being at work may give us another size, represented by, say, 1.75 cm length. And because one size can, and does, replace another, it is really distinct from the Essence of the (given particular) crystal -- so we say in substance-accident metaphysics -- because the Essence remains the same. This is the model.

But is the substitution of one particular length by another really accompanied by a constancy of that which has these sizes successively? Not necessarily. It is certainly plausible that there is no such constancy, or at least not in all cases :  While the crystal grows, the chance of dislocations and other irregularities to appear, changing the crystal's symmetry, increases, as also the chance of incoming and accepted (by the crystal lattice) chemical impurities. That is, it indicates that the so-called 'Essence' of the crystal does not remain the same during the succession of states the crystal goes through (we have considered the space group symmetry PLUS chemical composition as a direct phenotypical consequence of the crystal's Essence). Also when we compare two simultaneously existing similar individual crystals of different size (representing different growth stages), we will see that their (internal) symmetry is not the same, in the sense that they show different deviations from the space group that is theoretically expected for them :  They show different 'defects'. And also their chemical composition is different, albeit only slightly so, because of (different) chemical 'impurities'. So individuals of, say, snow crystals can have different sizes and even different shapes as a result of (varying) growing conditions, but these same conditions cause (later to be called) internal defects and chemical impurities, transforming the 'Essence' into another (formally different) 'Essence'. So we cannot say anymore that the crystal has  accidentally  changed (here :  altered its size or shape), because such a way of speaking suggests that  ' the crystal itself '  has remained the same ('only its size or shape has changed'). In fact the only things we see in a crystal (and in any being whatsoever) are features, features like space group symmetry, chemical composition, size shape, etc. And each one of them (one more so than the others) seems to vary in time (with respect to one particular crystal individual) as well as in space (several co-existing crystal individuals being observed simultaneously).
So when we describe a given patch of reality, we list such features and report their varying in time and space. And probably the World only consists of features, not only epistemologically, but also ontologically. We then preliminarily 'attach' these features to something that 'carries' them, that 'has' them, but that is only a matter of idealizing or modelling. It could be that any given feature is carried by the total but specific collection of features minus that given feature. But byintroducing a special collection of features, we are back to ontologically interpreting the structure of predication :

Socrates ( = total specific collection of features, minus the feature whiteness)  is  white  (where  ' white '  means that the feature  whiteness  is possessed or carried by something).

That, say a length of (say) 1.70 cm is just a feature (signified in such a way that the term  ' length of 1.70 cm '  belongs in the Category of Quantity) is evident from the fact that it can in principle also be attributed to some  aggregate  (as contrasted with a genuine single (fully dressed) substance.

If I see things correctly, our way of thinking, as it is anchored in language and logic, could represent a necessary a priori tool that makes knowledge possible in the first place. This is asserted in the philosophy of Kant. However,I prefer that these linguistic and logical structures are a posteriori acquired during biological evolution of (types of) living beings, in order to be able to deal with, and survive in, their environment, enabling such a being to make predictions, which are not necessarily well-deduced, but which are in most cases useful for survival.
So the mentioned structures are models, that become refined in metaphysics and natural science to illuminate certain aspects and relations between observed features.

Counter argument against the above described reality structure.
The considerations about growing crystals and their nonexistent Essence picture crystals, but by the same reasons also organisms, as totally mixed up in their environment. In fact so much (mixed up) that the distinction of crystal and its environment, or organism and its environment, become artificial and arbitrary distinctions. There are no things with identities of their own, because they do not have an 'own', they do not have an Essence.
However, the simple fact of the existence of self-conscious beings strongly opposes this view.
It may be that material things have turned out to be in a stronger degree interwoven with each other than was conceded by earlier generations of researchers, but this stronger degree of interwoveness does not necessarily destroy the possiblity of existing Essences. This is evident from the fact that we indeed can successfully classify natural crystals (doing it on the basis of space group symmetry and chemical composition), and that we can classify and recognize (albeit often with great difficulty) species of organisms. Moreover, natural science must necessarily presuppose a Species-Individuum Structure in material things, and consequently the possibility of exact repetition of some formal content (on which presupposition is based its fundamental method, induction, that is, generalization of individual findings). And although this presupposition allowes it to be weakened (and natural science then still be possible) in the sense that the repetition is never exact, it must be very close to it, in virtue of the fact that natural science has shown itself to be perfectly possible.
So although Essences are perhaps not sharply delineated features, they are nevertheless present. Their more or less vague boundaries being perhaps characteristic to them. Conceding the existence of such Essences makes it possible to speak of 'defects', of 'impurities', of 'essential features', of 'accidental features', of 'perturbations', etc., etc. And it has indeed (for example in crystallography and mineralogy) proven meaningful and fruitful to speak about them. In organisms we can meaningfully speak of diseases of the body and of accidental change of such a body, thereby presupposing the existence of some Essence in every living being. A very strong sign of the presence of an Essence in each organism is the possession by them of DNA, carrying characteristic information, not to mention the possession of self-consciousness by some of them.
So we stick to the substance-accident metaphysics, but, as we have already realized earlier, such a metaphysics (as any other metaphysics) is no more than a model of the ontological constitution of things. It is a stylized picture of this constitution.

When accepting a revised (as is accomplished on this website) substance-accident metaphysics, we must now turn to the following question :  What are the right logical entities that are precisely fit for the process of coming to know a reality which has an ontological structure according to such substance-accident metaphysics?
Logic studies the most basic types of formal entities and formal structures (for example by classifying them) that are necessarily involved in knowing, i.e. in knowing the objects of these formal entities and structures. The latter are concepts, propositions, arguments, and the like. Their common property is that they are such as to have an object. This object comes to be known to the human knower through these concepts, propositions, arguments, etc. So Logic studies concepts, propositions, arguments, and the like. It does, however, not study particular concepts, propositions and arguments as to their content which they happen to have. For instance, it does not ask what the analysed content of the concept  ' mouse '  is. This should be dealt with by natural science (biology). It asks what a concept as concept is, and also what a proposition as proposition is, etc. And this results in a classification, not of biological concepts say, but of fundamental  types  of concepts, fundamental  types  of propositions, and fundamental  types  of arguments. So Logic will find the very basic cognitive structures. And it is to be expected that these basic cognitive structures reflect correspondingly very basic strucures of their objects (that is, the things to be known). And what then is this basic structure? Or, said, differently, what does  basic structure  of the object mean? It cannot mean anything else than its  ontological  structure or composition, because that structure is truly basic.
So the logical entities as cognitive structures must, in order to be able to fulfill their function, be geared to the ontological structure of their objects.
Above we have conceded that the ontological structure of objects must be according to the model as presented by an amended substance-accident metaphysics. So we can now ask the question as to what type of Logic (because many types have been proposed) is precisely such as to provide the logical entities which are geared to such an ontological structure?
As far as I can see such a Logic must be of the type of an  intentional Logic,  because its logical entities are truly intentional :  they are nothing but (natural) signs, and their whole being consists in intending something other than themselves. As such they are formal signs. Only in this way the knower is directly connected with the thing to be known. And only in such a position he can come to know that thing. By means of the formal sign he has, as it were, internalized the thing. Modern mathematical Logic, on the other hand, is not so constituted. Its logical entities are not formal signs (and where they happen to be such, they are not interpreted as being such signs) but only instrumental signs. The structure of such instrumental signs are supposed to be isomorphic with that segment of reality of which they are about. This means that such signs are in fact real structures that are supposed to be similar to other structures (the latter being the ones that one wants to know). In this way a direct connection of the knower with the thing known is not established. In fact the knower comes to know the instrumental sign, but cannot reach further to apprehend the very thing that he wants to know. So is mathematical Logic then not a Logic after all? Indeed it is not. But what then is it? Well, it is a very interesting and important theory of all possible  relations .  We can also say that it is a mathematical discipline which investigates very basic possible formal structures in the light of a special mathematical function which is called the truth function, because of its analogy with the truth function present in the evaluation of knowledge.
It is Intential Logic which claims that it can provide (that is, disclose) truly formal signs, signs that establish a direct contact between the knower and the thing known.
The next document investigates whether there indeed are such formal signs. And only if we can make this plausible, we can start with a systematic exposition of Intentional Logic.

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To continue click HERE  for further study of Logic, Part II.

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